Question

Complete the square to solve the following quadratic equation: x^2−6x−20=20 .(1 point) Responses x=10 , x=−4 x equals 10 , x equals negative 4 x=−10 , x=4 x equals negative 10 , x equals 4 x=52 , x=−46 x equals 52 , x equals negative 46 x=3+210−−√, 3−210−−√

Answer 1

To solve the quadratic equation \( x^2 - 6x - 20 = 20 \) by completing the square, let's follow the steps:

1. **Move all terms to one side of the equation**:

\[
x^2 - 6x - 20 - 20 = 0
\]

This simplifies to:

\[
x^2 - 6x - 40 = 0
\]

2. **Isolate the quadratic and linear terms**:

\[
x^2 - 6x = 40
\]

3. **Complete the square**:

To complete the square, take the coefficient of \( x \) (which is \(-6\)), divide it by 2, and square it:

\[
\left( \frac{-6}{2} \right)^2 = 9
\]

Add this square to both sides:

\[
x^2 - 6x + 9 = 40 + 9
\]

This simplifies to:

\[
(x - 3)^2 = 49
\]

4. **Take the square root of both sides**:

\[
x - 3 = \pm 7
\]

5. **Solve for \( x \)**:

This gives us two equations:

\[
x - 3 = 7 \quad \Rightarrow \quad x = 10
\]

\[
x - 3 = -7 \quad \Rightarrow \quad x = -4
\]

Thus, the solution to the quadratic equation is:

\[
x = 10 \quad \text{and} \quad x = -4
\]

The correct response is:

**x equals 10, x equals negative 4**.