Complete the square to solve the following quadratic equation: x^2−6x−20=20

Question

.(1 point)
Responses

A) x=3+2√10, 3−2√10
x equals 3 plus 2 Start Root 10 End Root comma 3 minus 2 Start Root 10 End Root

B) x=52, x=−46
x equals 52 , x equals negative 46

C) x=−10, x=4
x equals negative 10 , x equals 4

D) x=10, x=−4

Answer 1

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

1. First, rearrange the equation so that all terms are on one side: \[ x^2 - 6x - 20 - 20 = 0 \] This simplifies to: \[ x^2 - 6x - 40 = 0 \]

2. Next, move the constant to the other side of the equation: \[ x^2 - 6x = 40 \]

3. To complete the square, take half of the coefficient of \( x \) (which is -6), square it, and add it to both sides: \[ \left(-\frac{6}{2}\right)^2 = (-3)^2 = 9 \] So, we add 9 to both sides: \[ x^2 - 6x + 9 = 40 + 9 \] This gives: \[ (x - 3)^2 = 49 \]

4. Now, take the square root of both sides: \[ x - 3 = \pm 7 \]

5. Solve for \( x \): \[ x - 3 = 7 \quad \text{or} \quad x - 3 = -7 \] Thus: \[ x = 10 \quad \text{or} \quad x = -4 \]

So, the solutions are \( x = 10 \) and \( x = -4 \).

The correct choice from the options provided is:

D) x = 10, x = -4