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+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−−√
𝑥
=
11
+
3
15
,

11
−
3
15
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
=
−
11
+
14
,

−
11
−
14
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root

Answer 1

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

1. **Start with the equation:**
\[
x^2 - 22x + 1 = 15
\]

2. **Subtract 15 from both sides to set the equation to zero:**
\[
x^2 - 22x + 1 - 15 = 0
\]
\[
x^2 - 22x - 14 = 0
\]

3. **Rearranging gives us:**
\[
x^2 - 22x = 14
\]

4. **Now, complete the square. We take half of the coefficient of \( x \), which is -22, giving us -11, and then square it:**
\[
(-11)^2 = 121
\]

5. **Add and subtract 121 on the left side:**
\[
x^2 - 22x + 121 - 121 = 14
\]
\[
(x - 11)^2 - 121 = 14
\]
\[
(x - 11)^2 = 14 + 121
\]
\[
(x - 11)^2 = 135
\]

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

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

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

After simplifying \( \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \), the solutions can be expressed as:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]

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