Question
x^2+16x-7=0 by completing the square
Answers
GPT-4o mini
To solve the quadratic equation \(x^2 + 16x - 7 = 0\) by completing the square, follow these steps:
1. **Move the constant term to the other side of the equation:**
\[
x^2 + 16x = 7
\]
2. **Complete the square:**
To do this, take the coefficient of \(x\), which is 16, divide it by 2 to get 8, and then square it to get 64. We will add and subtract 64 on the left side:
\[
x^2 + 16x + 64 - 64 = 7
\]
Rearranging, we have:
\[
(x + 8)^2 - 64 = 7
\]
3. **Move \(-64\) to the other side:**
\[
(x + 8)^2 = 7 + 64
\]
\[
(x + 8)^2 = 71
\]
4. **Take the square root of both sides:**
\[
x + 8 = \pm \sqrt{71}
\]
5. **Isolate \(x\):**
\[
x = -8 \pm \sqrt{71}
\]
So the solutions to the equation \(x^2 + 16x - 7 = 0\) are:
\[
x = -8 + \sqrt{71} \quad \text{and} \quad x = -8 - \sqrt{71}
\]
1. **Move the constant term to the other side of the equation:**
\[
x^2 + 16x = 7
\]
2. **Complete the square:**
To do this, take the coefficient of \(x\), which is 16, divide it by 2 to get 8, and then square it to get 64. We will add and subtract 64 on the left side:
\[
x^2 + 16x + 64 - 64 = 7
\]
Rearranging, we have:
\[
(x + 8)^2 - 64 = 7
\]
3. **Move \(-64\) to the other side:**
\[
(x + 8)^2 = 7 + 64
\]
\[
(x + 8)^2 = 71
\]
4. **Take the square root of both sides:**
\[
x + 8 = \pm \sqrt{71}
\]
5. **Isolate \(x\):**
\[
x = -8 \pm \sqrt{71}
\]
So the solutions to the equation \(x^2 + 16x - 7 = 0\) are:
\[
x = -8 + \sqrt{71} \quad \text{and} \quad x = -8 - \sqrt{71}
\]