Question

Complete the square to solve the following quadratic equation: x² - 22x + 1 = 15. (1 point)
x=-113√√15,
- 11 - 3√√15
x = 11+ √√√14, 11 – √√14
x = −11+ √√14, - 11 – √√14
x = 11+3√/15, 11 – 3√√/15

Answer 1

Let's complete the square to solve the quadratic equation \( x^2 - 22x + 1 = 15 \).

1. First, we can rewrite the equation in standard form:
\[
x^2 - 22x + 1 - 15 = 0
\]
which simplifies to:
\[
x^2 - 22x - 14 = 0
\]

2. Next, we want to complete the square. We take the coefficient of \( x \), which is \(-22\), divide it by 2 to get \(-11\), and then square it:
\[
(-11)^2 = 121
\]

3. We then add and subtract this square inside the equation:
\[
x^2 - 22x + 121 - 121 - 14 = 0
\]
which can be rearranged to:
\[
(x - 11)^2 - 135 = 0
\]

4. Now, we isolate the square:
\[
(x - 11)^2 = 135
\]

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

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

7. We can simplify \( \sqrt{135} \):
\[
\sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15}
\]
Thus, we have:
\[
x = 11 \pm 3\sqrt{15}
\]

The solutions are:
\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]

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