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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Please solve by using completing the square method thank you!
![### Problem 16: Solving Quadratic Equations by Completing the Square
**Objective:**
Solve the equation \( x^2 - 6x = 8 \) by completing the square.
**Steps for Completing the Square:**
1. **Original Equation:**
\[
x^2 - 6x = 8
\]
2. **Move the Constant Term to the Other Side:**
\[
x^2 - 6x - 8 = 0
\]
3. **Add and Subtract the Square of Half the Coefficient of \(x\):**
- Take half of the coefficient of \(x\) (which is -6), square it, and add and subtract this value inside the equation.
- Half of -6 is -3, and \((-3)^2 = 9\).
4. **Rewrite the Equation:**
\[
x^2 - 6x + 9 = 8 + 9
\]
5. **Simplify Both Sides:**
\[
(x-3)^2 = 17
\]
6. **Solve for \(x\):**
- Take the square root of both sides:
\[
x - 3 = \pm \sqrt{17}
\]
- Solve for \(x\):
\[
x = 3 \pm \sqrt{17}
\]
**Conclusion:**
The solutions for the equation \( x^2 - 6x = 8 \) are \( x = 3 + \sqrt{17} \) and \( x = 3 - \sqrt{17} \).
**Explanation:**
Completing the square involves turning a quadratic expression into a perfect square trinomial. This process makes it easier to solve quadratics that are not easily factorable, providing a pathway to finding solutions by using the square root method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3456819-528d-42b3-b255-e8873e975f3d%2F61cbaea3-9755-41e9-b91d-5f3b6f4e9098%2Fqzlchf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 16: Solving Quadratic Equations by Completing the Square
**Objective:**
Solve the equation \( x^2 - 6x = 8 \) by completing the square.
**Steps for Completing the Square:**
1. **Original Equation:**
\[
x^2 - 6x = 8
\]
2. **Move the Constant Term to the Other Side:**
\[
x^2 - 6x - 8 = 0
\]
3. **Add and Subtract the Square of Half the Coefficient of \(x\):**
- Take half of the coefficient of \(x\) (which is -6), square it, and add and subtract this value inside the equation.
- Half of -6 is -3, and \((-3)^2 = 9\).
4. **Rewrite the Equation:**
\[
x^2 - 6x + 9 = 8 + 9
\]
5. **Simplify Both Sides:**
\[
(x-3)^2 = 17
\]
6. **Solve for \(x\):**
- Take the square root of both sides:
\[
x - 3 = \pm \sqrt{17}
\]
- Solve for \(x\):
\[
x = 3 \pm \sqrt{17}
\]
**Conclusion:**
The solutions for the equation \( x^2 - 6x = 8 \) are \( x = 3 + \sqrt{17} \) and \( x = 3 - \sqrt{17} \).
**Explanation:**
Completing the square involves turning a quadratic expression into a perfect square trinomial. This process makes it easier to solve quadratics that are not easily factorable, providing a pathway to finding solutions by using the square root method.
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