Tugas mtk soal 2
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The document provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. It asks to: 1) Solve quadratic equations by factoring. 2) Solve quadratic equations by completing the square. 3) Solve quadratic equations using the quadratic formula. 4) Determine the value of p that makes a given quadratic equation have equal roots.
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Tugas mtk soal 2
1. The document provides solutions to several quadratic equations using factoring, completing the square, and the quadratic formula.
2. The equations are solved step-by-step showing the work. Common solutions included factoring the equation into two binomial factors and setting each factor equal to zero, or using the quadratic formula.
3. Additional context is provided about solving quadratic equations with different methods like completing the square or using the standard formula.
Tugas 1
1. The document is a math worksheet containing word problems to solve quadratic equations. It asks students to solve equations by factoring, completing the square, and using the quadratic formula.
2. Some example equations included x2 + 5x - 50 = 0, 2x2 + 3x + 1 = 0, and x2 - 15x + 30 = 0.
3. The last question asks students to determine the value of P such that the equation (p + 3)x3 + 3x - 4 = 0 has two equal roots.
Tugas 2
The document contains examples of solving quadratic equations by factorizing. It provides 5 examples in the first section of factorizing quadratic equations into their roots. The second section contains 2 examples of solving quadratic equations by completing the square. The third section provides 3 examples of solving quadratic equations using the quadratic formula. The final example solves a quadratic equation for an unknown parameter p.
Tugas matik ii
The document provides examples of factorizing quadratic equations and completing the square to solve quadratic equations. It discusses:
1) Factoring quadratic expressions like x2 + 5x - 50, x2 + 3x, and 2x2 + 3x + 1.
2) Completing the square to solve equations like x2 + 5x + 4 = 0 and 2x2 - 14x + 12 = 0.
3) Using the quadratic formula to solve equations like x2 - 15x + 30 = 0, x2 + 8x - 20 = 0, and x2 + 3x = 0.
4) Setting the discriminant equal to 0 to find the value of p that makes the equation
Substitution
The document provides examples of simplifying monomials, binomials, and trinomials using substitution. It works through problems involving substitution into expressions with variables raised to various powers and combinations of terms.
Simultaneous equations - Vedic Maths Training Classes Pune - Bhushan 9370571465
The document solves three systems of simultaneous equations. The first system is x + 2y = 10 and 2x + 3y = 18, which is solved to get x = 6 and y = 2. The second system is x + 3y = 11 and 2x + 3y = 13, which is solved to get x = 2 and y = 3. The third system is 4x + 5y = 14 and 3x + 3y = 9, which is solved to get x = 1 and y = 2.
Squaring a binomial
The document provides steps for squaring binomial expressions. It shows how to write the square of terms like (2x - 2y) by expanding it out to 4x^2 - 8xy + 4y^2. It then works through several examples of squaring binomials with different variables and coefficients like (x + 9), (10x - 1), and [(a + b) + c].
Algebra review trashketball
1) The slope of the line through (2, 5) and (-1, -7) is -3.
2) The solutions for x2 - 5x = 9 are x = 3, x = -3.
3) The solution for 3x + 7 < 5x - 10 in interval notation is (-∞, 2).
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Tugas mtk soal 2
1. The document provides solutions to several quadratic equations using factoring, completing the square, and the quadratic formula.
2. The equations are solved step-by-step showing the work. Common solutions included factoring the equation into two binomial factors and setting each factor equal to zero, or using the quadratic formula.
3. Additional context is provided about solving quadratic equations with different methods like completing the square or using the standard formula.
Tugas 1
1. The document is a math worksheet containing word problems to solve quadratic equations. It asks students to solve equations by factoring, completing the square, and using the quadratic formula.
2. Some example equations included x2 + 5x - 50 = 0, 2x2 + 3x + 1 = 0, and x2 - 15x + 30 = 0.
3. The last question asks students to determine the value of P such that the equation (p + 3)x3 + 3x - 4 = 0 has two equal roots.
Tugas 2
The document contains examples of solving quadratic equations by factorizing. It provides 5 examples in the first section of factorizing quadratic equations into their roots. The second section contains 2 examples of solving quadratic equations by completing the square. The third section provides 3 examples of solving quadratic equations using the quadratic formula. The final example solves a quadratic equation for an unknown parameter p.
Tugas matik ii
The document provides examples of factorizing quadratic equations and completing the square to solve quadratic equations. It discusses:
1) Factoring quadratic expressions like x2 + 5x - 50, x2 + 3x, and 2x2 + 3x + 1.
2) Completing the square to solve equations like x2 + 5x + 4 = 0 and 2x2 - 14x + 12 = 0.
3) Using the quadratic formula to solve equations like x2 - 15x + 30 = 0, x2 + 8x - 20 = 0, and x2 + 3x = 0.
4) Setting the discriminant equal to 0 to find the value of p that makes the equation
Substitution
The document provides examples of simplifying monomials, binomials, and trinomials using substitution. It works through problems involving substitution into expressions with variables raised to various powers and combinations of terms.
Simultaneous equations - Vedic Maths Training Classes Pune - Bhushan 9370571465
The document solves three systems of simultaneous equations. The first system is x + 2y = 10 and 2x + 3y = 18, which is solved to get x = 6 and y = 2. The second system is x + 3y = 11 and 2x + 3y = 13, which is solved to get x = 2 and y = 3. The third system is 4x + 5y = 14 and 3x + 3y = 9, which is solved to get x = 1 and y = 2.
Squaring a binomial
The document provides steps for squaring binomial expressions. It shows how to write the square of terms like (2x - 2y) by expanding it out to 4x^2 - 8xy + 4y^2. It then works through several examples of squaring binomials with different variables and coefficients like (x + 9), (10x - 1), and [(a + b) + c].
Algebra review trashketball
1) The slope of the line through (2, 5) and (-1, -7) is -3.
2) The solutions for x2 - 5x = 9 are x = 3, x = -3.
3) The solution for 3x + 7 < 5x - 10 in interval notation is (-∞, 2).
Tugas kelompok mtk soal 2
1. The document provides solutions to 5 equations:
2. Equations A, B, C are solved by factorizing the expressions.
3. Equations D and E are solved using the quadratic formula to find the two roots of each equation.
Squaring a binomial
The document provides step-by-step workings for squaring binomial expressions. It shows how to square terms like (2x - 2y) by expanding it to 4x^2 - 8xy + 4y^2. It then works through squaring more binomials with varying coefficients like (x + 9), (10x - 1), and composite expressions like [(a + b) + c]. Each is expanded using the formula (a + b)^2 = a^2 + 2ab + b^2.
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This document provides examples and explanations of operations and concepts involving polynomial and rational expressions. It begins with examples of factoring polynomials and using the factored form to evaluate expressions. It then covers topics such as combining like terms in rational expressions, multiplying and dividing rational expressions using factoring, simplifying complex fractions, and rationalizing denominators involving radicals. The document aims to demonstrate techniques for working with polynomials and rational expressions through step-by-step examples and explanations of related concepts.
Tugas ii (2)
This document contains information about a group assignment in Indonesian, including:
1. The name of the group leader and members.
2. Details of the assignment which involves factorizing quadratic equations, completing the square to put equations in the form x^2=p, and using the ABC formula.
3. Steps provided as examples to solve parts of the assignment involving factorizing, completing the square, and using the ABC formula.
Tugas ii (2)
1. The document lists the members of group b including the group leader and tasks.
2. It provides examples of factorizing quadratic equations such as x^2 + 5x - 50 = 0 into (x - 5)(x + 10) = 0.
3. It explains completing the square to write quadratic equations in the form of x^2 = p such as (x - 7)^2 = -14 + 12 for x^2 - 14x + 12 = 0.
4. It uses the quadratic formula to solve equations such as determining the value of p that makes the equation (p + 3)x^2 + 3x - 4 = 0 have two equal roots.
Tugas matematika menemukan konsep persamaan kuadrat
1. The document contains examples of solving quadratic equations by factorizing, completing the square, and using the quadratic formula.
2. The examples include finding the area of land by setting up and solving a quadratic equation, finding the volume of objects using quadratic equations, and determining the roots of various quadratic equations.
3. The key steps shown are factorizing, completing the square, setting the discriminant equal to zero, and using the quadratic formula to solve for the roots of the equations.
1.4 complex numbers t
This document discusses complex numbers. It provides examples of solving quadratic equations using complex numbers, multiplying and dividing complex numbers, and working with powers of i. It also includes exercises involving combining, expanding, dividing, and simplifying complex number expressions. Key points covered include using i=√-1, FOIL method for multiplication, conjugate multiplication, and the cyclic pattern of powers of i.
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This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
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The document contains examples of simple first-degree equations with:
- No parentheses or denominators
- Terms grouped together
- Parentheses
The equations are solved for the variable x, with the solutions provided.
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This document discusses solving quadratic equations by factoring. It begins by defining a quadratic equation in standard form and explaining the zero factor property. It then provides examples of solving quadratic equations through factoring and setting each factor equal to zero. Finally, it demonstrates solving word problems by setting up and solving the resulting quadratic equation.
แบบฝึกหัดการคูณจำนวนเต็ม
The document contains examples of multiplication exercises for students to practice. It includes exercises asking students to write out multiplication as repeated addition, fill in multiplication problems, calculate various multiplication problems involving positives and negatives, and substitute values into expressions involving variables. The exercises increase in complexity, starting with basic multiplication and moving towards more advanced problems combining multiplication, subtraction and variables.
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2) Using the quadratic formula.
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1. The document contains solutions to numerical methods problems involving roots, derivatives, approximations of fractions in binary, and matrix factorization.
2. For problem 1, the student computes derivatives of a function f(x) up to the 9th derivative to solve for roots of a polynomial.
3. For problem 2, the student uses binary approximations to find fractional representations of 1/3, 8/15, and 1/10 + 1/5 - 1/6.
4. For problem 3, the student proves a property of infinite geometric series and converts the decimal 8/7 to binary.
5. For problem 4, the student factors a coefficient matrix into upper and lower triangular matrices.
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1. This document discusses calculus formulas for derivatives of common functions including exponential, logarithmic, trigonometric, and implicit functions. It provides the derivative formulas and works through examples of finding derivatives of various functions.
2. Several examples are worked through, applying the formulas to find the derivatives of functions like y = ecos5x, y = (e4x - e5x)4, and implicit functions like x3 + y4 = 0.
3. The document concludes by providing the basic derivative formulas for inverse trigonometric functions and working through an example of finding the derivative of y = arc sin (5 + x2).
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The document discusses the zero factor theorem and provides examples of using it to solve quadratic equations. The zero factor theorem states that if p and q are algebraic expressions, then pq = 0 if and only if p = 0 or q = 0. This means a quadratic equation can be solved by factoring it into two linear factors and setting each factor equal to zero. Five examples are provided that show factoring quadratic equations, applying the zero factor theorem to set the factors equal to zero, and solving for the roots.
Extracting the roots
The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.
Tugas 2
The document contains examples of solving quadratic equations by factorizing. It provides 5 examples in the first section of factorizing quadratic equations into their roots. The second section contains 2 examples of solving quadratic equations by completing the square. The third section provides 3 examples of solving quadratic equations using the quadratic formula. The final section derives the discriminant of another quadratic equation to solve for the variable p.
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T ugaa kelompok mtk soal 2
1. The document provides solutions to 5 equations:
2. Equations A, B, C are solved by factorizing the expressions.
3. Equations D, E are solved using the quadratic formula to find the two roots of each equation.
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Tugas kelompok mtk soal 2
1. The document provides solutions to 5 equations:
2. Equations A, B, C are solved by factorizing the expressions.
3. Equations D and E are solved using the quadratic formula to find the two roots of each equation.
Squaring a binomial
The document provides step-by-step workings for squaring binomial expressions. It shows how to square terms like (2x - 2y) by expanding it to 4x^2 - 8xy + 4y^2. It then works through squaring more binomials with varying coefficients like (x + 9), (10x - 1), and composite expressions like [(a + b) + c]. Each is expanded using the formula (a + b)^2 = a^2 + 2ab + b^2.
1.2 algebraic expressions t
This document provides examples and explanations of operations and concepts involving polynomial and rational expressions. It begins with examples of factoring polynomials and using the factored form to evaluate expressions. It then covers topics such as combining like terms in rational expressions, multiplying and dividing rational expressions using factoring, simplifying complex fractions, and rationalizing denominators involving radicals. The document aims to demonstrate techniques for working with polynomials and rational expressions through step-by-step examples and explanations of related concepts.
Tugas ii (2)
This document contains information about a group assignment in Indonesian, including:
1. The name of the group leader and members.
2. Details of the assignment which involves factorizing quadratic equations, completing the square to put equations in the form x^2=p, and using the ABC formula.
3. Steps provided as examples to solve parts of the assignment involving factorizing, completing the square, and using the ABC formula.
Tugas ii (2)
1. The document lists the members of group b including the group leader and tasks.
2. It provides examples of factorizing quadratic equations such as x^2 + 5x - 50 = 0 into (x - 5)(x + 10) = 0.
3. It explains completing the square to write quadratic equations in the form of x^2 = p such as (x - 7)^2 = -14 + 12 for x^2 - 14x + 12 = 0.
4. It uses the quadratic formula to solve equations such as determining the value of p that makes the equation (p + 3)x^2 + 3x - 4 = 0 have two equal roots.
Tugas matematika menemukan konsep persamaan kuadrat
1. The document contains examples of solving quadratic equations by factorizing, completing the square, and using the quadratic formula.
2. The examples include finding the area of land by setting up and solving a quadratic equation, finding the volume of objects using quadratic equations, and determining the roots of various quadratic equations.
3. The key steps shown are factorizing, completing the square, setting the discriminant equal to zero, and using the quadratic formula to solve for the roots of the equations.
1.4 complex numbers t
This document discusses complex numbers. It provides examples of solving quadratic equations using complex numbers, multiplying and dividing complex numbers, and working with powers of i. It also includes exercises involving combining, expanding, dividing, and simplifying complex number expressions. Key points covered include using i=√-1, FOIL method for multiplication, conjugate multiplication, and the cyclic pattern of powers of i.
1 4 homework
This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
Ecuaciones
The document contains examples of simple first-degree equations with:
- No parentheses or denominators
- Terms grouped together
- Parentheses
The equations are solved for the variable x, with the solutions provided.
Lesson 5.3 honors
This document discusses solving quadratic equations by factoring. It begins by defining a quadratic equation in standard form and explaining the zero factor property. It then provides examples of solving quadratic equations through factoring and setting each factor equal to zero. Finally, it demonstrates solving word problems by setting up and solving the resulting quadratic equation.
แบบฝึกหัดการคูณจำนวนเต็ม
The document contains examples of multiplication exercises for students to practice. It includes exercises asking students to write out multiplication as repeated addition, fill in multiplication problems, calculate various multiplication problems involving positives and negatives, and substitute values into expressions involving variables. The exercises increase in complexity, starting with basic multiplication and moving towards more advanced problems combining multiplication, subtraction and variables.
Ecs lineales
This document contains 26 equations to solve for the variable x. The equations involve adding, subtracting, multiplying and dividing terms with x. Students are instructed to copy the equations into their notebook and solve for x in each case. The solutions provided range from simple integers to more complex fractional values of x.
Day 7 review (2)
The document contains mathematical expressions and equations involving variables and functions. Several examples are shown of finding zeros of polynomial functions by factoring or using the rational root theorem. Graphs are plotted of various functions and key features are identified such as relative maxima, minima, intervals of increasing and decreasing behavior.
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Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
Quadratic eq and discriminant
The document discusses the quadratic formula and how to use it to solve quadratic equations. It introduces the quadratic formula as x = (-b ±√(b2 - 4ac))/2a and explains how to use it to solve equations by plugging in the a, b, and c coefficients. It also defines the discriminant b2 - 4ac and explains what types of roots (real, complex, rational, irrational) an equation will have depending on whether the discriminant is positive, negative, or zero. Several examples are worked through to demonstrate applying the quadratic formula.
Sbma 4603 numerical methods Assignment
1. The document contains solutions to numerical methods problems involving roots, derivatives, approximations of fractions in binary, and matrix factorization.
2. For problem 1, the student computes derivatives of a function f(x) up to the 9th derivative to solve for roots of a polynomial.
3. For problem 2, the student uses binary approximations to find fractional representations of 1/3, 8/15, and 1/10 + 1/5 - 1/6.
4. For problem 3, the student proves a property of infinite geometric series and converts the decimal 8/7 to binary.
5. For problem 4, the student factors a coefficient matrix into upper and lower triangular matrices.
Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4
1. This document discusses calculus formulas for derivatives of common functions including exponential, logarithmic, trigonometric, and implicit functions. It provides the derivative formulas and works through examples of finding derivatives of various functions.
2. Several examples are worked through, applying the formulas to find the derivatives of functions like y = ecos5x, y = (e4x - e5x)4, and implicit functions like x3 + y4 = 0.
3. The document concludes by providing the basic derivative formulas for inverse trigonometric functions and working through an example of finding the derivative of y = arc sin (5 + x2).
Factoring
The document discusses the zero factor theorem and provides examples of using it to solve quadratic equations. The zero factor theorem states that if p and q are algebraic expressions, then pq = 0 if and only if p = 0 or q = 0. This means a quadratic equation can be solved by factoring it into two linear factors and setting each factor equal to zero. Five examples are provided that show factoring quadratic equations, applying the zero factor theorem to set the factors equal to zero, and solving for the roots.
Extracting the roots
The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.
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Solving Equations Transformable to Quadratic Equation Including Rational Alge...
Tugas mtk soal 2
- 1. 1. Selesaikan persamaan kuadrat di bawah dengan cara memfaktorkan a. x2 + 5x - 50 = 0 b. x2 + 3x = 0 c. x2 - 4 = 0 d. 2x2 + 3x + 1 = 0 e. 3x2 + 5x - 2 = 0 Jawab: a. x2 + 5x - 50 = 0 x2 + 10x – 5x – 50 = 0 (x2 + 10x) – (5x – 50) = 0 x(x + 10) – 5(x + 10) = 0 (x – 5) (x + 10) = 0 x = 5 atau x = -10 b. x2 + 3x = 0 x2 + 2x + x = 0 (x2 + 2x) + x = 0 x(x + 3) = 0 x = -3 c. x2 - 4 = 0 x2 + 2x – 2x – 4 = 0 (x2 +2x) – (2x – 4) = 0 x(x + 2) – 2(x + 2) = 0 (x – 2) (x + 2) = 0 x = 2 atau x = -2 d. 2x2 + 3x + 1 = 0 2x2 + 2x + x + 1 = 0 (2x2 + 2x) + (x + 1) = 0 2x(x +1) + (x + 1) = 0 (2x + 1) (x +1) = 0 x = - atau x = -1
- 2. e. 3x2 + 5x - 2 = 0 3x2 – x + 6x -2 = 0 (3x2 – x) + (6x – 2) = 0 x(3x – x) + 2(3x –1) = 0 (x + 2) (3x – 1) = 0 x = -2 atau x = 2. Selesaikan persamaan kuadrat berikut ini dengan melengkapkan kuadrat sempurna a. x2 + 5x +4 = 0 b. 2x2 – 14x + 12 = 0 Jawab: a. x2 + 5x +4 = 0 x2 + 5x +4 – 4 = 0 -4 x2 + 5x + ( )2 = ( )2 – 4 (x + 5)2 = ( 5)2 – 4 2 (x + 5) = ± -4 2 X =- 5± 2 = -2,5 ± -4 -4 = -2,5± = -2,5 ± = -2,5± 1,5 x = -2,5 + 1,5 = -1, atau x = -2,5 – 1,5 = -4 b. 2x2 – 14x + 12 = 0 X2 – 7x + 6 = 0 X2 – 7x + 6 – 6 = 0 – 6 X2 – 7x + ( 7)2 = ( .7)2 – 6 (x + 7)2 = ( 7)2 – 6 (x + 7) = ± X=- 7± 2 -6 2 -6
- 3. 2 = -3,5 ± -6 = -3,5± = -3,5 ± = -3,5± 2,5 X = -3,5 + 2,5 = -1 atau X = -3,5 – 2,5 = -6 3. Selesaikan persamaan kuadrat berikut ini dengan rumus abc a. x2 - 15x + 30 = 0 Jawab: X1,2 b. x2+ 8x - 20 = 0 Jawab: X1,2 X1 = =2 X2 = = = -10
- 4. c. x 2 + 3x = 0 Jawab: X1,2 X1 X2 = = = -3 4. Tentukan p agar persamaan kuadrat (p+3)x2 + 3x - 4 = 0 mempunyai dua akar sama. Jawab: (p+3)x2 + 3x - 4 = 0 D = b2 – 4ac = 32 – 4(p+3). – 4 = 9 – 4p – 12. – 4 = 9+ 16p + 48 -16p = 9 + 48 -16p = 57 P=