Questions and Answers
What does lesson 1 refer to?
Solving quadratics by factoring
Which equations are true? (Select all that apply)
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?
Signup and view all the answers
What is the factored form of the expression x^2 − 8x − 48?
Signup and view all the answers
What is the factored form of the expression x^2 − 2x − 63?
Signup and view all the answers
What are the zeros of the function f(x) = x^2 − 10x + 21? (Select all that apply)
Signup and view all the answers
What are the solutions to the equation 3(x + 2)(x − 9) = 0? (Select all that apply)
Signup and view all the answers
Solve the equation 40 = 2x^2 + 2x by factoring. What are the solutions?
Signup and view all the answers
What are the zeros of the function f(x) = x^2 − 36?
Signup and view all the answers
What mistakes, if any, did Enrique make when solving an equation?
Signup and view all the answers
Solve the equation x^2 + 21x + 110 = 0 by factoring. What are the solutions?
Signup and view all the answers
What is the factored form of the expression 3x^2 + 6x − 24?
Signup and view all the answers
What is the factored form of the expression 9x^2 + 42x + 49?
Signup and view all the answers
What are the solutions to the equation (x − 4)(x + 2) = 0? (Select all that apply)
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?
Signup and view all the answers
What is the factored form of the expression x^2 − 2x − 15?
Signup and view all the answers
Solve the equation 25b^2 − 64 = 0 by factoring. What are the solutions?
Signup and view all the answers
What is the factored form of the expression x^2 − 7x − 18?
Signup and view all the answers
What is the factored form of the expression 36x^2 − 49y^2?
Signup and view all the answers
Solve the equation −2x^2− 2x + 40 = 0 by factoring. What are the solutions?
Signup and view all the answers
What does lesson 2 refer to?
Signup and view all the answers
What are the solutions of the equation x^2 + 15 = 79? (Select all that apply)
Signup and view all the answers
What are the solutions of the equation x^2 = 25/36?
Signup and view all the answers
What are the solutions, if any, of the equation x^2 + 38 = 16?
Signup and view all the answers
What are the solutions of the equation (x - 9)^2 = 25? (Select all that apply)
Signup and view all the answers
What are the solutions, if any, of the equation (x + 49)^2 = 24?
Signup and view all the answers
What mistake, if any, did Ellen make when solving the equation x^2 − 5 = 59?
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?
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?
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)
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)
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.