Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![### Solving Quadratic Equations by Factoring
To solve quadratic equations by factoring, follow the steps outlined below. Use the practice equation provided to test your understanding.
#### Problem
Find all real solutions of the equation by factoring. (Enter your answers as a comma-separated list.)
\[ x^2 - 12x + 32 = 0 \]
#### Solution
1. **Identify the quadratic equation in standard form (ax^2 + bx + c = 0)**
\[ x^2 - 12x + 32 = 0 \]
2. **Factor the quadratic expression on the left-hand side**
- Looking for two numbers that multiply to 32 (constant term) and add up to -12 (coefficient of x). Those numbers are -4 and -8:
\[ x^2 - 4x - 8x + 32 = 0 \]
- Factor by grouping:
\[ x(x - 4) - 8(x - 4) = 0 \]
\[ (x - 4)(x - 8) = 0 \]
3. **Set each factor to zero and solve for x**
\[ x - 4 = 0 \implies x = 4 \]
\[ x - 8 = 0 \implies x = 8 \]
4. **Combine the solutions in a comma-separated list**
\[ x = 4, 8 \]
#### Box Representation for Answer Input
\[ x = \boxed{\ \ \ \ \ \ \ \ \ \ \ \ \ } \]
##### Additional Resources:
- [Factoring Quadratics](https://exampleeducation.com/factoring_quadratics): Learn more about factoring methods.
- [Quadratic Equations](https://exampleeducation.com/quadratic_equations): Comprehensive guide on solving quadratic equations.
By practicing these steps, you can solve any quadratic equation by factoring. Be sure to check each step for accuracy and comprehend how each factor influences the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ad8a3e2-d44f-406e-ad02-79162c0dd845%2Fc9a85a1f-5003-4585-af7b-451d37d79b73%2Ftxidu5g_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Quadratic Equations by Factoring
To solve quadratic equations by factoring, follow the steps outlined below. Use the practice equation provided to test your understanding.
#### Problem
Find all real solutions of the equation by factoring. (Enter your answers as a comma-separated list.)
\[ x^2 - 12x + 32 = 0 \]
#### Solution
1. **Identify the quadratic equation in standard form (ax^2 + bx + c = 0)**
\[ x^2 - 12x + 32 = 0 \]
2. **Factor the quadratic expression on the left-hand side**
- Looking for two numbers that multiply to 32 (constant term) and add up to -12 (coefficient of x). Those numbers are -4 and -8:
\[ x^2 - 4x - 8x + 32 = 0 \]
- Factor by grouping:
\[ x(x - 4) - 8(x - 4) = 0 \]
\[ (x - 4)(x - 8) = 0 \]
3. **Set each factor to zero and solve for x**
\[ x - 4 = 0 \implies x = 4 \]
\[ x - 8 = 0 \implies x = 8 \]
4. **Combine the solutions in a comma-separated list**
\[ x = 4, 8 \]
#### Box Representation for Answer Input
\[ x = \boxed{\ \ \ \ \ \ \ \ \ \ \ \ \ } \]
##### Additional Resources:
- [Factoring Quadratics](https://exampleeducation.com/factoring_quadratics): Learn more about factoring methods.
- [Quadratic Equations](https://exampleeducation.com/quadratic_equations): Comprehensive guide on solving quadratic equations.
By practicing these steps, you can solve any quadratic equation by factoring. Be sure to check each step for accuracy and comprehend how each factor influences the solution.
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