Question

Complete the square to solve the following quadratic equation: x2−22x+1=15
𝑥
2
−
22
𝑥
+
1
=
15
.(1 point)
Responses

x=11+14−−√, 11−14−−√
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
=
−
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
=
−
11
+
3
15
,

−
11
−
3
15
x equals negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

x=11+315−−√, 11−315−−√

Answer 1

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

1. **Rearrange the equation**:
\[
x^2 - 22x + 1 - 15 = 0
\]
Simplifying gives us:
\[
x^2 - 22x - 14 = 0
\]

2. **Move the constant term to the other side**:
\[
x^2 - 22x = 14
\]

3. **Complete the square**:
- Take the coefficient of \(x\) (which is \(-22\)), divide it by 2, and square it:
\[
\left(-\frac{22}{2}\right)^2 = (-11)^2 = 121
\]

- Add and subtract \(121\) on the left-hand side:
\[
x^2 - 22x + 121 - 121 = 14
\]
This simplifies to:
\[
(x - 11)^2 - 121 = 14
\]

4. **Rewrite the equation**:
\[
(x - 11)^2 = 14 + 121
\]
Which simplifies to:
\[
(x - 11)^2 = 135
\]

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

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

Now we can rewrite \(\sqrt{135}\) as \(\sqrt{9 \times 15} = 3\sqrt{15}\), giving us:

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

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

So the correct response is:
\[
x = 11 + 3\sqrt{15}, \quad 11 - 3\sqrt{15}
\]