Question

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

Answer 1

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

1. **Rearrange the equation**: Move 15 to the left side:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]

2. **Isolate the quadratic and linear terms**:
\[
x^2 - 22x = 14
\]

3. **Complete the square**:
- Take half of the coefficient of \(x\) (which is -22), square it, and add it to both sides:
- Half of -22 is -11.
- Squaring -11 gives 121.
- Add 121 to both sides:
\[
x^2 - 22x + 121 = 14 + 121
\]
This simplifies to:
\[
(x - 11)^2 = 135
\]

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

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

6. **Simplify \(\sqrt{135}\)**:
\[
\sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15}
\]
Thus, the solutions can be written as:
\[
x = 11 \pm 3\sqrt{15}
\]

So the final solutions to the equation \( x^2 - 22x + 1 = 15 \) are:
\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]