Question

Complete the square to solve the following quadratic equation: x2−22x+1=15 .(1 point) Responses x=−11+14−−√, −11−14−−√ x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root x=11+315−−√, 11−315−−√ x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 End Root x=11+14−−√, 11−14−−√ x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 End Root x=−11+315−−√, −11−315−−√

Answer 1

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

1. **Move the constant to the other side**:
\[
x^2 - 22x + 1 - 15 = 0
\]
\[
x^2 - 22x - 14 = 0
\]

2. **Add 14 to both sides**:
\[
x^2 - 22x = 14
\]

3. **Complete the square**:
Take half of the coefficient of \( x \) (which is \(-22\)), square it, and add it to both sides. Half of \(-22\) is \(-11\), and \((-11)^2 = 121\).

\[
x^2 - 22x + 121 = 14 + 121
\]
\[
x^2 - 22x + 121 = 135
\]

4. **Rewrite the left side as a square**:
\[
(x - 11)^2 = 135
\]

5. **Take the square root of both sides**:
\[
x - 11 = \pm \sqrt{135}
\]

6. **Solve for \( x \)**:
\[
x = 11 \pm \sqrt{135}
\]

7. **Simplify \( \sqrt{135} \)**:
Notice that:
\[
\sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15}
\]

So we have:
\[
x = 11 \pm 3\sqrt{15}
\]

Thus, the final solutions are:
\[
x = 11 + 3\sqrt{15}, \qquad x = 11 - 3\sqrt{15}
\]

**Answer**: \( x=11 + 3\sqrt{15}, \, 11 - 3\sqrt{15} \) is the correct choice.