Complete the square to solve the following quadratic equation: x2−22x+1=15

Question

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