Complete the square to solve the following quadratic equation: x2+2x−8=0

Question

𝑥
2
+
2
𝑥
−
8
=
0
.(1 point)
Responses

x=−10
𝑥
=
−
10
, x=8
𝑥
=
8
x equals negative 10 , x equals 8

x=2
𝑥
=
2
, x=−4
𝑥
=
−
4
x equals 2 , x equals negative 4

x=10
𝑥
=
10
, x=−8
𝑥
=
−
8
x equals 10 , x equals negative 8

x=−2
𝑥
=
−
2
, x=4
𝑥
=
4
x equals negative 2 , x equals 4

Answer 1

To solve the quadratic equation \( x^2 + 2x - 8 = 0 \) by completing the square, follow these steps:

1. Start with the equation: \[ x^2 + 2x - 8 = 0 \]

2. Move the constant term to the other side: \[ x^2 + 2x = 8 \]

3. To complete the square, take half of the coefficient of \( x \) (which is 2), square it, and add it to both sides. Half of 2 is 1, and squaring it gives \( 1^2 = 1 \). \[ x^2 + 2x + 1 = 8 + 1 \] This simplifies to: \[ (x + 1)^2 = 9 \]

4. Now take the square root of both sides: \[ x + 1 = \pm 3 \]

5. Solve for \( x \) in both cases:

   • For the positive case: \[ x + 1 = 3 \implies x = 3 - 1 = 2 \]
   • For the negative case: \[ x + 1 = -3 \implies x = -3 - 1 = -4 \]

The solutions to the equation are: \[ x = 2 \quad \text{and} \quad x = -4 \]

Thus, the correct response is: x = 2, x = -4 (x equals 2, x equals negative 4).