Solve each problem by completing the square (must show work) 5. 2x² + 9 = 6. 8x 2x²7x - 30 = 0
Solve each problem by completing the square (must show work) 5. 2x² + 9 = 6. 8x 2x²7x - 30 = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Solving Quadratic Equations by Completing the Square**
**Instructions:** Solve each problem by completing the square. (Must show work)
---
**Problem 5:**
\[ 2x^2 + 9 = 8x \]
**Problem 6:**
\[ 2x^2 - 7x - 30 = 0 \]
---
### Explanation:
Completing the square is a method used to solve quadratic equations. This technique involves creating a perfect square trinomial from the quadratic equation, which can then be easily solved.
#### **Steps to Complete the Square:**
1. **Rearrange the Equation:** Ensure the equation is in the form \( ax^2 + bx + c = 0 \).
2. **Move the Constant Term:** If not needed on one side, move it to the other side of the equation.
3. **Normalize the Coefficient of \( x^2 \):** If \( a \neq 1 \), divide the entire equation by \( a \).
4. **Find the Term to Complete the Square:** Take half of the coefficient of \( x \), square it, and add this square on both sides of the equation.
5. **Form the Perfect Square Trinomial:** The equation should now be a perfect square on one side.
6. **Solve for \( x \):** Use the square root method and solve for \( x \).
---
By following these steps systematically, you will be able to solve the given quadratic equations by completing the square.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87177919-7483-46f8-b1a9-edbc0a3ed313%2Fb40270b6-77c2-4333-a564-abdfe4544a73%2F6net17_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Quadratic Equations by Completing the Square**
**Instructions:** Solve each problem by completing the square. (Must show work)
---
**Problem 5:**
\[ 2x^2 + 9 = 8x \]
**Problem 6:**
\[ 2x^2 - 7x - 30 = 0 \]
---
### Explanation:
Completing the square is a method used to solve quadratic equations. This technique involves creating a perfect square trinomial from the quadratic equation, which can then be easily solved.
#### **Steps to Complete the Square:**
1. **Rearrange the Equation:** Ensure the equation is in the form \( ax^2 + bx + c = 0 \).
2. **Move the Constant Term:** If not needed on one side, move it to the other side of the equation.
3. **Normalize the Coefficient of \( x^2 \):** If \( a \neq 1 \), divide the entire equation by \( a \).
4. **Find the Term to Complete the Square:** Take half of the coefficient of \( x \), square it, and add this square on both sides of the equation.
5. **Form the Perfect Square Trinomial:** The equation should now be a perfect square on one side.
6. **Solve for \( x \):** Use the square root method and solve for \( x \).
---
By following these steps systematically, you will be able to solve the given quadratic equations by completing the square.
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