Question

Complete the square to solve the following quadratic equation: x^2 −22x+1=15
.(1 point)

Answer 1

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:

1. **Move the constant to the other side of the equation:**

\[
x^2 - 22x + 1 - 15 = 0
\]
\[
x^2 - 22x - 14 = 0
\]

2. **Rearrange the equation for completion of the square:**

\[
x^2 - 22x = 14
\]

3. **Find the coefficient of \( x \), divide by 2, and square it:**

The coefficient of \( x \) is \(-22\). Half of \(-22\) is \(-11\), and squaring it gives:

\[
(-11)^2 = 121
\]

4. **Add and subtract this square inside the equation:**

\[
x^2 - 22x + 121 - 121 = 14
\]
\[
(x - 11)^2 - 121 = 14
\]

5. **Now, move 121 to the other side:**

\[
(x - 11)^2 = 14 + 121
\]
\[
(x - 11)^2 = 135
\]

6. **Take the square root of both sides:**

\[
x - 11 = \pm \sqrt{135}
\]

This can be simplified since \( \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \):

\[
x - 11 = \pm 3\sqrt{15}
\]

7. **Solve for \( x \):**

\[
x = 11 \pm 3\sqrt{15}
\]

Thus, the solutions are:

\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]