Question
Rewrite the equation by completing the square. \( x^{2}-6 x-16=0 \) \( (x+\square)^{2}=\square \)
Ask by Mathis Hammond. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
(x - 3)^2 = 25
\]
Solution
To rewrite the equation \( x^{2} - 6x - 16 = 0 \) by completing the square, follow these steps:
1. **Move the constant term to the other side:**
\[
x^{2} - 6x = 16
\]
2. **Complete the square for the quadratic expression on the left:**
- Take half of the coefficient of \( x \), which is \( -6 \), resulting in \( -3 \).
- Square this value: \( (-3)^2 = 9 \).
- Add this square to both sides of the equation:
\[
x^{2} - 6x + 9 = 16 + 9
\]
3. **Simplify both sides:**
\[
(x - 3)^2 = 25
\]
So, the equation rewritten by completing the square is:
\[
(x - 3)^2 = 25
\]
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Simplify this solution Beyond the Answer
To rewrite the equation by completing the square, start with the quadratic equation \( x^{2} - 6x - 16 = 0 \). Move the constant to the other side: \( x^{2} - 6x = 16 \). Now, to complete the square, take half of the coefficient of \( x \) (which is -6), square it, and add it to both sides. Half of -6 is -3, and squaring it gives 9. Then, we get: \( x^{2} - 6x + 9 = 16 + 9 \) This simplifies to: \( (x - 3)^{2} = 25 \) So, the final equation is: \( (x - 3)^{2} = 25 \)
