Complete the square to solve the following quadratic equation: x2+2x−8=0 .(1 point) Responses x=10, x=−8 x equals 10 , x equals negative 8 x=−2, x=4 x equals negative 2 , x equals 4 x=2, x=−4 x equals 2 , x equals negative 4 x=−10, x=8
To complete the square, we want to rewrite the quadratic equation in the form: (x + a)^2 = b.
To do this, take half of the coefficient of x and square it, which is 2/2 = 1. Add 1 to both sides of the equation:
x^2 + 2x − 8 + 1 = 1
x^2 + 2x − 7 = 1
Now, factor the left side of the equation:
(x + 1)^2 = 9
Taking the square root of both sides:
x + 1 = ±3
Subtracting 1 from both sides:
x = -1 ± 3
This gives us two possible solutions:
x = -1 + 3 = 2
x = -1 - 3 = -4
Therefore, the correct solution is x = 2, x = -4.
To complete the square to solve a quadratic equation, follow these steps:
1. Start with the equation in the form: ax^2 + bx + c = 0.
In this case, the given equation is: x^2 + 2x - 8 = 0.
2. Move the constant term to the other side of the equation:
x^2 + 2x = 8.
3. Take half of the coefficient of 'x', square it, and add it to both sides of the equation:
x^2 + 2x + 1 = 8 + 1.
The coefficient of 'x' is 2. Half of 2 is 1, and 1 squared is 1. Therefore, we can write: 1^2 = 1.
So, the equation becomes: x^2 + 2x + 1 = 9.
4. Rewrite the left side of the equation as a perfect square:
(x + 1)^2 = 9.
5. Take the square root of both sides of the equation:
√(x + 1)^2 = ±√9.
√(x + 1)^2 = ±3.
Simplifying gives: x + 1 = ±3.
6. Solve for 'x' by subtracting 1 from both sides of the equation:
x + 1 - 1 = ±3 - 1.
x = -1 ± 3.
So, there are two possible solutions:
x = -1 + 3 = 2.
x = -1 - 3 = -4.
Therefore, the correct answer is x = 2, x = -4.
To complete the square to solve the quadratic equation x^2 + 2x - 8 = 0, follow these steps:
1. Move the constant term (-8) to the other side of the equation by adding 8 to both sides:
x^2 + 2x = 8
2. Take half of the coefficient of the x-term (2), square it, and add it to both sides of the equation:
x^2 + 2x + (2/2)^2 = 8 + (2/2)^2
x^2 + 2x + 1 = 8 + 1
3. Simplify both sides of the equation:
x^2 + 2x + 1 = 9
4. Rewrite the left side of the equation as a perfect square:
(x + 1)^2 = 9
5. Take the square root of both sides of the equation:
√(x + 1)^2 = √9
6. Solve for x by setting up two separate equations, one with a positive square root and one with a negative square root:
x + 1 = 3 or x + 1 = -3
7. Solve each equation for x:
For x + 1 = 3:
x = 3 - 1
x = 2
For x + 1 = -3:
x = -3 - 1
x = -4
So, the solutions to the quadratic equation x^2 + 2x - 8 = 0, after completing the square, are x = 2 and x = -4.