The document discusses the discriminant of a quadratic equation and what it reveals about the nature of the roots. The discriminant, Δ, is calculated as b2 - 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, there are two equal real roots. If Δ < 0, there are no real roots. Several examples of finding the discriminant of equations and describing the roots are shown. The values of k that would result in equal or non-real roots for particular equations are also determined. Finally, the value of a for which the line y = ax is tangent to a given circle is found by setting the discriminant of the equation equal to 0.
The document discusses the discriminant of a quadratic equation and what it reveals about the nature of the roots. The discriminant, Δ, is calculated as b2 - 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, there are two equal real roots. If Δ < 0, there are no real roots. Several examples of finding the discriminant of equations and describing the roots are shown. The values of k that would result in equal or non-real roots for particular equations are also determined. Finally, the value of a for which the line y = ax is tangent to a given circle is found by setting the discriminant of the equation equal to 0.
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
11X1 T10 07 sum and product of roots (2010)Nigel Simmons
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
The document provides instructions for simplifying algebraic fractions. It states that one should always factorize the expression first before cancelling terms. Several worked examples are provided that show the steps to (1) create a common denominator, (2) identify the difference between the old and new denominators, and (3) multiply the numerator by this difference when factorizing.
O autor compartilha fotos de ninhos de pássaros construídos por um oleiro chamado João-de-Barro. Ele ficou surpreso com a reação de familiares e amigos ao compartilhar esta curiosidade diária e decidiu publicar as fotos online para que outras pessoas, especialmente aqueles com crianças, também pudessem apreciá-las.
Servis & Cia es una empresa joven dedicada a la venta, asistencia técnica, calibración y reparación de balanzas y básculas para el comercio, la industria y la agroganadería. Ofrece una variedad de productos como pesas para ganado, cerdos y accesorios como indicadores electrónicos, así como servicios de hibridación, mantenimiento y asistencia técnica para asegurar el correcto funcionamiento de los equipos.
The document discusses the discriminant of a quadratic equation and what it reveals about the nature of the roots. The discriminant, Δ, is calculated as b2 - 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, there are two equal real roots. If Δ < 0, there are no real roots. Several examples of finding the discriminant of equations and describing the roots are shown. The values of k that would result in equal or non-real roots for particular equations are also determined. Finally, the value of a for which the line y = ax is tangent to a given circle is found by setting the discriminant of the equation equal to 0.
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
11X1 T10 07 sum and product of roots (2010)Nigel Simmons
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
The document provides instructions for simplifying algebraic fractions. It states that one should always factorize the expression first before cancelling terms. Several worked examples are provided that show the steps to (1) create a common denominator, (2) identify the difference between the old and new denominators, and (3) multiply the numerator by this difference when factorizing.
O autor compartilha fotos de ninhos de pássaros construídos por um oleiro chamado João-de-Barro. Ele ficou surpreso com a reação de familiares e amigos ao compartilhar esta curiosidade diária e decidiu publicar as fotos online para que outras pessoas, especialmente aqueles com crianças, também pudessem apreciá-las.
Servis & Cia es una empresa joven dedicada a la venta, asistencia técnica, calibración y reparación de balanzas y básculas para el comercio, la industria y la agroganadería. Ofrece una variedad de productos como pesas para ganado, cerdos y accesorios como indicadores electrónicos, así como servicios de hibridación, mantenimiento y asistencia técnica para asegurar el correcto funcionamiento de los equipos.
The document discusses the discriminant of a quadratic equation and what it reveals about the nature of the roots. It gives the following information:
- The discriminant (Δ) tells us whether the roots are real or imaginary.
- If Δ > 0, there are two different real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots.
- If Δ is a perfect square, the roots are rational numbers.
- Several examples of quadratic equations are worked through to demonstrate applying the discriminant.
- Conditions are identified for quadratic equations to have equal, unreal or real roots based on the value of Δ.
11X1 T10 07 sum & product of roots (2011)Nigel Simmons
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
11 x1 t10 07 sum & product of roots (2012)Nigel Simmons
The document discusses the sum and product of roots for a quadratic equation ax^2 + bx + c = 0. It shows that:
- The sum of the roots (α + β) is equal to -b/a
- The product of the roots (αβ) is equal to c/a
It also provides examples of forming quadratic equations given specific roots, and calculating properties of the roots for the equation 2x^2 - 3x - 1 = 0.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
11 x1 t01 08 completing the square (2013)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows examples of solving equations in the form (i) x^2 + bx + c = 0, (ii) ax^2 + bx + c = 0, and (iii) x^2 - 6x + 6 = 0. The method involves grouping like terms and factorizing the equation into the form (x + p)^2 = q to extract the solutions.
11X1 T01 09 completing the square (2011)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows how to complete the square for a general quadratic equation of the form ax2 + bx + c = 0 by grouping like terms and factorizing into a perfect square form. Examples are worked through, including solving the specific equations x2 + 6x - 7 = 0, x2 - 6x + 6 = 0.
11X1 t01 08 completing the square (2012)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows how to complete the square for a general quadratic equation of the form ax2 + bx + c = 0 by grouping like terms and factorizing into a perfect square form. Examples are worked through, including solving the specific equations x2 + 6x - 7 = 0, x2 - 6x + 6 = 0.
The document discusses relationships between the coefficients and roots of polynomial equations. For quadratic equations ax2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. For cubic equations ax3 + bx2 + cx + d = 0, the sum of the roots is -b/a and the formulas extend to higher order polynomials. Examples are given to demonstrate calculating expressions involving the roots of a cubic equation from its coefficients.
11 x1 t10 07 sum & product of roots (2013)Nigel Simmons
The document discusses properties of the sum and product of roots of quadratic equations. It shows that:
- The sum of the roots α and β is equal to -b/a
- The product of the roots is equal to c/a
- Examples are given of forming quadratic equations with given roots and calculating properties like the sum and product of roots for specific equations.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type of curve, it gives the standard form of the equation and notes on identifying features like intercepts, vertices, and the overall shape of the graph. Sketching curves involves finding intercepts and using a table of values to plot points if needed.
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type of curve, it gives the standard form of the equation and notes on identifying features like intercepts, vertices, and asymptotes to help sketch the graph. It emphasizes using factorized forms, intercepts, and tables of values to determine a curve's shape.
The document discusses how to sketch graphs based on their equations. It provides the following key points:
- Numbers on axes must be evenly spaced. The y-intercept occurs when x=0 and the x-intercept occurs when y=0.
- Common curves include straight lines, parabolas, cubics, and higher order polynomials. Parabolas have x-intercepts found by solving the equation for where it equals 0.
- Higher order polynomials become flatter at the base and steeper on the sides as the power increases. Hyperbolas can be defined by equations like y=1/x or xy=1.
Once intercepts are found, curves can be sketched by
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type it provides the standard form of the equation and notes on identifying features like intercepts, vertices, and behavior as powers increase. It emphasizes using standard forms, intercepts, and factoring to determine a curve's shape.
This document discusses quadratic equations and how to determine their roots. It provides:
1) The general form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
2) Methods for determining the roots of a quadratic equation, including factorization, completing the square, and using the quadratic formula.
3) How to determine the type of roots (real/complex, equal/distinct) based on the discriminant b2 - 4ac.
Modul penggunaan kalkulator sainstifik sebagai ABM dalam MatematikNorsyazana Kamarudin
This document provides information about discriminants of quadratic equations. It defines quadratic equations and explains that the discriminant, which is b^2 - 4ac, provides information about the number and type of roots. A positive discriminant indicates two real roots, a zero discriminant indicates one real root, and a negative discriminant indicates no real roots. Examples of solving quadratic equations with a scientific calculator are provided. Worksheets ask students to determine the type of roots and solutions for different quadratic equations using the discriminant and with or without a calculator.
The document discusses the nature of roots of a quadratic equation and how it relates to the discriminant. It begins by recalling the quadratic formula and defining the discriminant. It then describes the three cases for the nature of roots based on the discriminant:
1) If the discriminant is greater than 0, there are two unequal real roots.
2) If the discriminant is equal to 0, there is one double real root.
3) If the discriminant is less than 0, there are no real roots.
Some examples are provided to illustrate each case. Finally, it summarizes that the value of the discriminant determines the nature of the roots, which also corresponds to the number of x-
The document discusses the signs of quadratic functions and how to determine if they are positive definite, negative definite, or indefinite based on the values of a, Δ, and k. An example problem demonstrates how to find the values of k that make the quadratic function kx2 - 6x + k positive definite by analyzing the signs of a and Δ. The solution is that k must be greater than 3.
The document discusses the signs of quadratic functions and how to determine if they are positive definite, negative definite, or indefinite based on the values of a, Δ, and k. An example problem demonstrates how to find the values of k that make the quadratic function kx2 - 6x + k positive definite by analyzing the signs of a and Δ. The solution is that k must be greater than 3.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses the discriminant of a quadratic equation and what it reveals about the nature of the roots. It gives the following information:
- The discriminant (Δ) tells us whether the roots are real or imaginary.
- If Δ > 0, there are two different real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots.
- If Δ is a perfect square, the roots are rational numbers.
- Several examples of quadratic equations are worked through to demonstrate applying the discriminant.
- Conditions are identified for quadratic equations to have equal, unreal or real roots based on the value of Δ.
11X1 T10 07 sum & product of roots (2011)Nigel Simmons
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
11 x1 t10 07 sum & product of roots (2012)Nigel Simmons
The document discusses the sum and product of roots for a quadratic equation ax^2 + bx + c = 0. It shows that:
- The sum of the roots (α + β) is equal to -b/a
- The product of the roots (αβ) is equal to c/a
It also provides examples of forming quadratic equations given specific roots, and calculating properties of the roots for the equation 2x^2 - 3x - 1 = 0.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
11 x1 t01 08 completing the square (2013)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows examples of solving equations in the form (i) x^2 + bx + c = 0, (ii) ax^2 + bx + c = 0, and (iii) x^2 - 6x + 6 = 0. The method involves grouping like terms and factorizing the equation into the form (x + p)^2 = q to extract the solutions.
11X1 T01 09 completing the square (2011)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows how to complete the square for a general quadratic equation of the form ax2 + bx + c = 0 by grouping like terms and factorizing into a perfect square form. Examples are worked through, including solving the specific equations x2 + 6x - 7 = 0, x2 - 6x + 6 = 0.
11X1 t01 08 completing the square (2012)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows how to complete the square for a general quadratic equation of the form ax2 + bx + c = 0 by grouping like terms and factorizing into a perfect square form. Examples are worked through, including solving the specific equations x2 + 6x - 7 = 0, x2 - 6x + 6 = 0.
The document discusses relationships between the coefficients and roots of polynomial equations. For quadratic equations ax2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. For cubic equations ax3 + bx2 + cx + d = 0, the sum of the roots is -b/a and the formulas extend to higher order polynomials. Examples are given to demonstrate calculating expressions involving the roots of a cubic equation from its coefficients.
11 x1 t10 07 sum & product of roots (2013)Nigel Simmons
The document discusses properties of the sum and product of roots of quadratic equations. It shows that:
- The sum of the roots α and β is equal to -b/a
- The product of the roots is equal to c/a
- Examples are given of forming quadratic equations with given roots and calculating properties like the sum and product of roots for specific equations.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type of curve, it gives the standard form of the equation and notes on identifying features like intercepts, vertices, and the overall shape of the graph. Sketching curves involves finding intercepts and using a table of values to plot points if needed.
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type of curve, it gives the standard form of the equation and notes on identifying features like intercepts, vertices, and asymptotes to help sketch the graph. It emphasizes using factorized forms, intercepts, and tables of values to determine a curve's shape.
The document discusses how to sketch graphs based on their equations. It provides the following key points:
- Numbers on axes must be evenly spaced. The y-intercept occurs when x=0 and the x-intercept occurs when y=0.
- Common curves include straight lines, parabolas, cubics, and higher order polynomials. Parabolas have x-intercepts found by solving the equation for where it equals 0.
- Higher order polynomials become flatter at the base and steeper on the sides as the power increases. Hyperbolas can be defined by equations like y=1/x or xy=1.
Once intercepts are found, curves can be sketched by
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type it provides the standard form of the equation and notes on identifying features like intercepts, vertices, and behavior as powers increase. It emphasizes using standard forms, intercepts, and factoring to determine a curve's shape.
This document discusses quadratic equations and how to determine their roots. It provides:
1) The general form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
2) Methods for determining the roots of a quadratic equation, including factorization, completing the square, and using the quadratic formula.
3) How to determine the type of roots (real/complex, equal/distinct) based on the discriminant b2 - 4ac.
Modul penggunaan kalkulator sainstifik sebagai ABM dalam MatematikNorsyazana Kamarudin
This document provides information about discriminants of quadratic equations. It defines quadratic equations and explains that the discriminant, which is b^2 - 4ac, provides information about the number and type of roots. A positive discriminant indicates two real roots, a zero discriminant indicates one real root, and a negative discriminant indicates no real roots. Examples of solving quadratic equations with a scientific calculator are provided. Worksheets ask students to determine the type of roots and solutions for different quadratic equations using the discriminant and with or without a calculator.
The document discusses the nature of roots of a quadratic equation and how it relates to the discriminant. It begins by recalling the quadratic formula and defining the discriminant. It then describes the three cases for the nature of roots based on the discriminant:
1) If the discriminant is greater than 0, there are two unequal real roots.
2) If the discriminant is equal to 0, there is one double real root.
3) If the discriminant is less than 0, there are no real roots.
Some examples are provided to illustrate each case. Finally, it summarizes that the value of the discriminant determines the nature of the roots, which also corresponds to the number of x-
The document discusses the signs of quadratic functions and how to determine if they are positive definite, negative definite, or indefinite based on the values of a, Δ, and k. An example problem demonstrates how to find the values of k that make the quadratic function kx2 - 6x + k positive definite by analyzing the signs of a and Δ. The solution is that k must be greater than 3.
The document discusses the signs of quadratic functions and how to determine if they are positive definite, negative definite, or indefinite based on the values of a, Δ, and k. An example problem demonstrates how to find the values of k that make the quadratic function kx2 - 6x + k positive definite by analyzing the signs of a and Δ. The solution is that k must be greater than 3.
Similar to 11 x1 t10 05 the discriminant (2012) (20)
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
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The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
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The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
3. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
4. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
5. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
6. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
7. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
8. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. (i ) Describe the roots of;
9. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. (i ) Describe the roots of;
a) 3x 2 5 x 9 0
10. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. (i ) Describe the roots of;
a) 3x 2 5 x 9 0
52 4 3 9
83 0
11. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. (i ) Describe the roots of;
a) 3x 2 5 x 9 0
52 4 3 9
83 0
no real roots
12. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. (i ) Describe the roots of;
a) 3x 2 5 x 9 0 b ) 2x 2 6 x 3 0
52 4 3 9
83 0
no real roots
13. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. (i ) Describe the roots of;
a) 3x 2 5 x 9 0 b ) 2x 2 6 x 3 0
52 4 3 9 62 4 2 3
83 0 60 0
no real roots
14. The Discriminant
b 2 4ac
The discriminant tells us whether the roots are rational or irrational
0 : two different real roots (cuts the x axis twice)
0 : two equal real roots (touches the x axis once)
0 : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. (i ) Describe the roots of;
a) 3x 2 5 x 9 0 b ) 2x 2 6 x 3 0
52 4 3 9 62 4 2 3
83 0 60 0
no real roots two different, real, irrational roots
15. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
16. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
equal roots occur when 0
17. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
equal roots occur when 0
i.e. 62 4k 0
18. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
equal roots occur when 0
i.e. 62 4k 0
36 4k 0
k 9
19. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
equal roots occur when 0
i.e. 62 4k 0
36 4k 0
k 9
b) x 2 4 x 2k 0 have unreal roots
20. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
equal roots occur when 0
i.e. 62 4k 0
36 4k 0
k 9
b) x 2 4 x 2k 0 have unreal roots
unreal roots occur when 0
21. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
equal roots occur when 0
i.e. 62 4k 0
36 4k 0
k 9
b) x 2 4 x 2k 0 have unreal roots
unreal roots occur when 0
i.e. 4 4 2k 0
2
22. (ii) Find the values of k which makes;
a ) x 2 6 x k 0 have equal roots
equal roots occur when 0
i.e. 62 4k 0
36 4k 0
k 9
b) x 2 4 x 2k 0 have unreal roots
unreal roots occur when 0
i.e. 4 4 2k 0
2
16 8k 0
k 2
24. c) kx 2 2 x 4k 0 have real roots
real roots occur when 0
25. c) kx 2 2 x 4k 0 have real roots
real roots occur when 0
i.e. 22 4 k 4k 0
26. c) kx 2 2 x 4k 0 have real roots
real roots occur when 0
i.e. 22 4 k 4k 0
4 16k 2 0
1
k
2
4
27. c) kx 2 2 x 4k 0 have real roots
real roots occur when 0
i.e. 22 4 k 4k 0
4 16k 2 0
1
k
2
4
1 1
k
2 2
28. c) kx 2 2 x 4k 0 have real roots
real roots occur when 0
i.e. 22 4 k 4k 0
4 16k 2 0
1
k
2
4
1 1
k
2 2
(iii ) For what value of a is the line y ax a tangent to
the circle x 2 y 2 20 x 10 y 100 0?
29. c) kx 2 2 x 4k 0 have real roots
real roots occur when 0
i.e. 22 4 k 4k 0
4 16k 2 0
1
k
2
4
1 1
k
2 2
(iii ) For what value of a is the line y ax a tangent to
the circle x 2 y 2 20 x 10 y 100 0?
x 2 a 2 x 2 20 x 10ax 100 0
30. c) kx 2 2 x 4k 0 have real roots
real roots occur when 0
i.e. 22 4 k 4k 0
4 16k 2 0
1
k
2
4
1 1
k
2 2
(iii ) For what value of a is the line y ax a tangent to
the circle x 2 y 2 20 x 10 y 100 0?
x 2 a 2 x 2 20 x 10ax 100 0
a 2
1 x 2 10 2 a x 100 0