Complete the square to solve the following quadratic equation: x^2 + 2x - 8 = 0.
A. x = -2, x = 4
B. x = 10, x = -8
C. x = -10, x = 8
D. x = 2, x = -4
The given quadratic equation is x^2 + 2x - 8 = 0.
To complete the square, we will add and subtract the square of half of the coefficient of x.
In this case, the coefficient of x is 2, so:
x^2 + 2x - 8 = 0
x^2 + 2x + (2/2)^2 - (2/2)^2 - 8 = 0
x^2 + 2x + 1 - 1 - 8 = 0
(x + 1)^2 - 9 = 0
(x + 1)^2 = 9
(x + 1) = ±√9
x + 1 = ±3
x = -1 ± 3
x = -1 + 3 = 2
x = -1 - 3 = -4
Therefore, the solutions to the quadratic equation x^2 + 2x - 8 = 0 are x = 2 and x = -4.
The correct answer is D. x = 2, x = -4.
To complete the square to solve the quadratic equation x^2 + 2x - 8 = 0, follow these steps:
Step 1: Move the constant term (-8) to the other side of the equation:
x^2 + 2x = 8
Step 2: Divide the coefficient of x (2) by 2, and then square the result: (2/2)^2 = 1
Add this value to both sides of the equation:
x^2 + 2x + 1 = 8 + 1
x^2 + 2x + 1 = 9
Step 3: Rewrite the left side of the equation as a perfect square trinomial by factoring:
(x + 1)^2 = 9
Step 4: Take the square root of both sides to solve for x:
√((x + 1)^2) = √9
x + 1 = ±3
Step 5: Solve for x by subtracting 1 from both sides:
x = -1 ± 3
Simplifying the solutions gives us:
x = -1 + 3 = 2
x = -1 - 3 = -4
Therefore, the solutions to the quadratic equation x^2 + 2x - 8 = 0 after completing the square are:
D. x = 2, x = -4.
To solve the quadratic equation x^2 + 2x - 8 = 0 by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation.
x^2 + 2x = 8
Step 2: Take half of the coefficient of the x-term (which is 2) and square it.
(2/2)^2 = 1
Step 3: Add the square from Step 2 to both sides of the equation.
x^2 + 2x + 1 = 9
Step 4: Rewrite the left side of the equation as a perfect square.
(x + 1)^2 = 9
Step 5: Take the square root of both sides of the equation.
√(x + 1)^2 = ±√9
Step 6: Solve for x by setting up two equations, one with the positive square root and one with the negative square root.
x + 1 = ±3
Step 7: Solve each equation for x.
x = -1 + 3 = 2
x = -1 - 3 = -4
Therefore, the solutions to the quadratic equation x^2 + 2x - 8 = 0 obtained by completing the square are x = 2 and x = -4.
So, the correct answer is option D: x = 2, x = -4.