Algebra: Solving Quadratics by Factoring

Questions and Answers

What does lesson 1 refer to?

Solving quadratics by factoring

Which equations are true? (Select all that apply)

9x^2 - 1 = (3x - 1)(3x + 1) (correct)

x^2 - 169 = (x - 13)(x + 13) (correct)

16x^2 - 64y^2 = (4x - 8y)(4x + 8y) (correct)

None of the above

What is the factored form of the expression x^2 − 22x + 121?

(x - 11)^2

What is the factored form of the expression 2m^3 − 26m^2 + 80m?

2m(m - 5)(m - 8) Signup and view all the answers

What is the factored form of the expression x^2 − 8x − 48?

(x + 4)(x - 12) Signup and view all the answers

What is the factored form of the expression x^2 − 2x − 63?

(x + 7)(x - 9) Signup and view all the answers

What are the zeros of the function f(x) = x^2 − 10x + 21? (Select all that apply)

3 Signup and view all the answers

What are the solutions to the equation 3(x + 2)(x − 9) = 0? (Select all that apply)

-2 Signup and view all the answers

Solve the equation 40 = 2x^2 + 2x by factoring. What are the solutions?

x = -5 and x = 4 Signup and view all the answers

What are the zeros of the function f(x) = x^2 − 36?

Zeros occur when x = 6 and x = -6 Signup and view all the answers

What mistakes, if any, did Enrique make when solving an equation?

Enrique used the difference of squares pattern incorrectly when factoring. Signup and view all the answers

Solve the equation x^2 + 21x + 110 = 0 by factoring. What are the solutions?

-10, -11 Signup and view all the answers

What is the factored form of the expression 3x^2 + 6x − 24?

3(x + 4)(x - 2) Signup and view all the answers

What is the factored form of the expression 9x^2 + 42x + 49?

(3x + 7)^2 Signup and view all the answers

What are the solutions to the equation (x − 4)(x + 2) = 0? (Select all that apply)

-2 Signup and view all the answers

What mistakes, if any, did Gemma make when finding the zeros of the function f(x) = 3x^2 − 6x − 45?

In step 4, when solving for x + 3 = 0, Gemma should have subtracted 3 from both sides of the equation. Signup and view all the answers

What is the factored form of the expression x^2 − 2x − 15?

(x + 3)(x - 5) Signup and view all the answers

Solve the equation 25b^2 − 64 = 0 by factoring. What are the solutions?

8/5, -8/5 Signup and view all the answers

What is the factored form of the expression x^2 − 7x − 18?

(x + 2)(x - 9) Signup and view all the answers

What is the factored form of the expression 36x^2 − 49y^2?

(6x + 7y)(6x - 7y) Signup and view all the answers

Solve the equation −2x^2− 2x + 40 = 0 by factoring. What are the solutions?

-5, 4 Signup and view all the answers

What does lesson 2 refer to?

Other methods for solving quadratics Signup and view all the answers

What are the solutions of the equation x^2 + 15 = 79? (Select all that apply)

8 Signup and view all the answers

What are the solutions of the equation x^2 = 25/36?

-5/6, 5/6 Signup and view all the answers

What are the solutions, if any, of the equation x^2 + 38 = 16?

This equation has no real solutions. Signup and view all the answers

What are the solutions of the equation (x - 9)^2 = 25? (Select all that apply)

14 Signup and view all the answers

What are the solutions, if any, of the equation (x + 49)^2 = 24?

x = −49 ± 2√6 Signup and view all the answers

What mistake, if any, did Ellen make when solving the equation x^2 − 5 = 59?

In line 4, she forgot to take the negative square root, because squaring both 8 and −8 equals 64. Signup and view all the answers

Solve the quadratic equation −4x^2 + 6x + 16 = 0 by completing the square. Which expression represents the correct solutions?

3 ± √73 / 4 Signup and view all the answers

Solve the quadratic equation −x^2 + 8x − 4 = 0 by completing the square. Which expression represents the correct solutions?

4 ± 2√3 Signup and view all the answers

What statements identify Nando's mistakes when he solved the quadratic equation 4x^2 − 24x − 16 = 0 by completing the square? (Select all that apply)

In step 7, his final answer should have been x = 3 ± √13. Signup and view all the answers

Solve the quadratic equation x^2 − 14x + 24 = 0 by completing the square. What are the solutions to the equation? (Select all that apply)

2 Signup and view all the answers

Study Notes

Lesson 1: Solving Quadratics by Factoring

Quadratic equations can be solved by factoring into products of binomials.
True equations include:
- ( 9x^2 - 1 = (3x - 1)(3x + 1) )
- ( x^2 - 169 = (x - 13)(x + 13) )
- ( 16x^2 - 64y^2 = (4x - 8y)(4x + 8y) )

Factored Forms of Quadratics

Factored form of ( x^2 - 22x + 121 ) is ( (x - 11)^2 ).
Factored form of ( 2m^3 - 26m^2 + 80m ) is ( 2m(m - 5)(m - 8) ).
Factored form of ( x^2 - 8x - 48 ) is ( (x + 4)(x - 12) ).
Factored form of ( x^2 - 2x - 63 ) is ( (x + 7)(x - 9) ).
Factored form of ( 3x^2 + 6x - 24 ) is ( 3(x + 4)(x - 2) ).
Factored form of ( 9x^2 + 42x + 49 ) is ( (3x + 7)^2 ).
Factored form of ( x^2 - 7x - 18 ) is ( (x + 2)(x - 9) ).
Factored form of ( 36x^2 - 49y^2 ) is ( (6x + 7y)(6x - 7y) ).

Zeros and Solutions

Zeros of ( f(x) = x^2 - 10x + 21 ) are ( 3 ) and ( 7 ).
Solutions to ( 3(x + 2)(x - 9) = 0 ) are ( -2 ) and ( 9 ).
Solutions to the equation ( 40 = 2x^2 + 2x ) are ( -5 ) and ( 4 ).
Zeros of ( f(x) = x^2 - 36 ) occur at ( x = 6 ) and ( x = -6 ).
Solutions to ( (x - 4)(x + 2) = 0 ) are ( -2 ) and ( 4 ).
Solutions to ( x^2 + 21x + 110 = 0 ) are ( -10 ) and ( -11 ).
Solutions of ( 25b^2 - 64 = 0 ) are ( \frac{8}{5} ) and ( -\frac{8}{5} ).

Common Mistakes

Mistakes can occur in applying factoring patterns correctly, such as the difference of squares.
Ensure to consider both positive and negative roots when solving equations.
Be cautious about dividing all parts of an equation equally when simplifying.

Lesson 2: Other Methods for Solving Quadratics

Squaring both sides can yield two solutions, such as ( (x - 9)^2 = 25 ) yielding ( 14 ) and ( 4 ).
Some equations may have no real solutions, for instance, ( x^2 + 38 = 16 ).

Completing the Square

Completing the square can provide solutions in a different form, such as:
- For ( -4x^2 + 6x + 16 = 0 ), solutions are ( 3 \pm \frac{\sqrt{73}}{4} ).
- For ( -x^2 + 8x - 4 = 0 ), solutions are ( 4 \pm 2\sqrt{3} ).
Notable mistakes in completing the square include not handling constants correctly or neglecting the negative square root.

Summary of Solutions

Common solutions to equations include ( 8, -8 ) for ( x^2 + 15 = 79 ) and ( -\frac{5}{6}, \frac{5}{6} ) for ( x^2 = \frac{25}{36} ).
The equation ( (x + 49)^2 = 24 ) simplifies to ( x = -49 \pm 2\sqrt{6} ).

Overall Strategies

Familiarize with different methods such as factoring, completing the square, and recognizing when an equation has no real solutions.
Practice deriving zeros, factoring, and correcting mistakes in approaches to enhance problem-solving skills.

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Description

This quiz is designed to test your understanding of solving quadratic equations by factoring. You will explore various true equations and their factored forms, as well as find zeros and solutions. Perfect for reinforcing key concepts in quadratic functions.