Question

Complete the square to solve the following quadratic equation: x^2−22x+1=15
.

Answer 1

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

1. **Move all terms to one side of the equation**:
\[
x^2 - 22x + 1 - 15 = 0
\]
Simplifying this, we get:
\[
x^2 - 22x - 14 = 0
\]

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

3. **Complete the square**:
To complete the square, we take the coefficient of \( x \) (which is \(-22\)), divide it by 2, and then square it:
\[
\left(-\frac{22}{2}\right)^2 = (-11)^2 = 121
\]
Now, add 121 to both sides:
\[
x^2 - 22x + 121 = 14 + 121
\]
This simplifies to:
\[
x^2 - 22x + 121 = 135
\]

4. **Rewrite the left side as a square**:
The left side is now a perfect square:
\[
(x - 11)^2 = 135
\]

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

6. **Solve for \( x \)**:
Adding 11 to both sides gives:
\[
x = 11 \pm \sqrt{135}
\]

7. **Simplify \( \sqrt{135} \)**:
We can simplify \( \sqrt{135} \) as:
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]
Thus, the solutions for \( x \) are:
\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]

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