Bca3010 computer oriented numerical methods
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Solving Quadratic Equations by Completing the Square
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two intercept form
The document discusses using the two intercept form to find the equation of a line given two points. It provides the two intercept form equation, where a and b are the x and y intercepts. It then works through three examples of finding the line equation using two points and the two intercept form. It lists additional practice problems and their solutions for finding line equations using two points and the two intercept form.
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https://tinyurl.com/y9muob6q
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https://tinyurl.com/ycjp8r7u
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The document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the form ax2 + bx + c, where a ≠ 0. The constants a, b, and c are the quadratic, linear, and constant coefficients.
2) There are three main methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula.
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Quadratic equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
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The document discusses solving quadratic equations using the quadratic formula. It defines the discriminant as the expression under the radical sign in the quadratic formula, and explains that the discriminant determines the number of real roots: a positive discriminant means two real roots, zero discriminant means one real root, and a negative discriminant means no real roots. Examples are provided to demonstrate solving quadratic equations using the quadratic formula and interpreting the discriminant.
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QUADRATIC EQUATIONS
This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.
Bca3010 computer oriented numerical methods
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
Bca3010 computer oriented numerical methods
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
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Geometric mean
Geometric means are the terms between non-consecutive terms of a geometric sequence. If A1, A2, .....An-1, An is a geometric sequence, then the numbers A2,.....,An-1 are the geometric means between A1 and An. Three examples are provided to illustrate how to find geometric means between given numbers: inserting geometric means between 1 and 81, between 8 and 512, and finding the single geometric mean between 5 and 8.
Strategic intervention materials on mathematics 2.0
This document provides teaching materials on solving quadratic equations by factoring for a mathematics class. It includes an overview of quadratic equations and their standard form. It then outlines least mastered skills and activities to practice identifying quadratic equations, rewriting them in standard form, factoring trinomials, and determining roots. Example problems and solutions are provided to demonstrate factoring trinomials and using factoring to solve quadratic equations. A practice problem asks students to solve a word problem involving a quadratic equation. Key terms and concepts are bolded. References for further reading are listed at the end.
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Solving Quadratic Equations
This document discusses how to solve quadratic equations by factoring, using the quadratic formula, and determining the number and type of roots using the discriminant. Key steps include factoring the equation if possible to set each factor equal to 0 and solve, plugging the coefficients a, b, and c into the quadratic formula if factoring is not possible, and using the discriminant b^2 - 4ac to determine the number and type of roots. Examples are provided to demonstrate each method.
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This document introduces methods for solving quadratic equations beyond factoring, including the square root property, completing the square, and the quadratic formula. It discusses how to determine the number and type of solutions based on the discriminant. The key steps are presented for solving quadratics, graphing quadratic functions as parabolas, and finding the domain and range. Piecewise-defined quadratic functions are also explained.
quadratic equation
The document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the form ax2 + bx + c, where a ≠ 0. The constants a, b, and c are the quadratic, linear, and constant coefficients.
2) There are three main methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula.
3) The discriminant, b2 - 4ac, determines the nature of the roots - two real roots if positive, one real root if zero, or two complex roots if negative.
Geometric series
This document provides information on calculating the sum of terms in a geometric series using formulas. It gives three examples of finding the sum of terms for different geometric sequences: (1) the sum of the first 12 terms of 4, 16, 64,... (2) the sum of the first 7 terms of -1, -5, -25, -125,... (3) the sum of the first 10 terms where the first term is 1/2 and the fourth term is 4. The key steps are to identify the initial term (A1), common ratio (r), and number of terms (n); then apply the formula Sn = A1(1 - rn)/(1 - r) to calculate the sum, where r
Quadratic equation
The document discusses different methods for solving quadratic equations. It explains that quadratic equations arise in various situations and fields of mathematics. Several methods are covered, including solving by square root property, factorization, completing the square, and using the quadratic formula. The quadratic formula provides the solutions to a quadratic equation in the form of ax2 + bx + c = 0 and depends on the discriminant to determine the number and type of solutions.
Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic Formula
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using the Quadratic Formula. It also discusses the steps in solving quadratic equations using the method of Quadratic Formula.
16.2 Solving by Factoring
This document provides an overview of solving quadratic equations by factoring. It begins with the standard form of a quadratic equation and explains the zero factor property. Examples are provided to demonstrate factoring quadratic equations and setting each factor equal to zero to solve. The steps for solving a quadratic equation by factoring are outlined. Additional examples demonstrate solving real world application problems involving quadratic equations.
Quadratic equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
Solving Equations Transformable to Quadratic Equation Including Rational Alge...
Provides examples and solutions on how to solve equations that is transformable to quadratic equation and rational algebraic equations.
Solving Linear Systems
The document provides examples of solving different types of linear systems of equations, including graphing, substitution, and addition methods. It demonstrates setting up and solving systems to find unknown variables from word problems involving gardeners and helpers earning different amounts. The final section defines a linear system as a set of two or more linear equations.
Mathematics Form 1-Chapter 5-6 Algebraic Expression Linear Equations KBSM of ...
This document summarizes a math chapter about algebraic expressions and linear equations. It covers topics like algebraic terms with multiple unknowns, multiplication and division of terms, and solving linear equations. It provides examples and exercises for students to practice the concepts. Key points introduced are the definitions of unknowns, coefficients, like and unlike terms, and how to perform operations and solve equations involving algebraic expressions.
16.6 Quadratic Formula & Discriminant
The document discusses solving quadratic equations using the quadratic formula. It defines the discriminant as the expression under the radical sign in the quadratic formula, and explains that the discriminant determines the number of real roots: a positive discriminant means two real roots, zero discriminant means one real root, and a negative discriminant means no real roots. Examples are provided to demonstrate solving quadratic equations using the quadratic formula and interpreting the discriminant.
Quadratic formula
The quadratic formula can be derived by completing the square on the standard form quadratic equation ax^2 + bx + c = 0. By completing the square, the equation can be written as (x + b/2a)^2 = (b^2 - 4ac)/4a^2, from which the quadratic formula -b ± √(b^2 - 4ac)/2a can be identified.
Solving Quadratic Equations
Provides example on how to solve quadratic equations using extracting square roots, factoring, completing the square and quadratic formula.
2.quadratic equations
This document contains 14 problems involving quadratic equations. The problems cover expressing quadratic equations in standard form, finding roots of quadratic equations, determining the range of values for constants in quadratic equations, and relating the roots and coefficients of quadratic equations. Sample solutions are provided for each problem.
QUADRATIC EQUATIONS
This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.
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Mathematics Form 1-Chapter 5-6 Algebraic Expression Linear Equations KBSM of ...
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Bca3010 computer oriented numerical methods
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
Bca3010 computer oriented numerical methods
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
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Send your semester & Specialization name to our mail id :
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rational equation transformable to quadratic equation.pptx
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