Question

Completing the Square & the Quadratic Formula Quick Check
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Question
Complete the square to solve the following quadratic equation: x2−22x+1=15.(1 point)
Responses

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+315−−√, −11−315−−√
x equals negative 11 plus 3 Start Root 15 End Root comma negative 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+14−−√, −11−14−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root
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Answer 1

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

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

2. **Rearrange and isolate the \( x^2 \) and \( x \) terms**:
\[
x^2 - 22x = 14
\]

3. **Complete the square**:
- Take half of the coefficient of \( x \) (which is -22), square it, and add 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 squared term**:
\[
(x - 11)^2 = 135
\]

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

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

So the solutions are:
\[
x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135}
\]

Now, simplifying \(\sqrt{135}\):
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]

Thus, we substitute back:
\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]

Therefore, the correct answer is:
\[
x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}
\]

Hence, the correct response from the options would be:
- **x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}**.