Sure, let's go through the steps to solve the equation [tex]\(-2x = x^2 - 6\)[/tex] by completing the square.
1. Rewrite the equation in standard quadratic form:
[tex]\[
x^2 + 2x - 6 = 0
\][/tex]
2. Complete the square:
- To complete the square, we need to convert the quadratic equation into the form [tex]\((x + c)^2 = d\)[/tex].
- Our quadratic term is [tex]\(x^2\)[/tex] and the linear term is [tex]\(2x\)[/tex].
- Take half of the coefficient of [tex]\(x\)[/tex] (which is 2), and square it.
[tex]\[
\left(\frac{2}{2}\right)^2 = 1
\][/tex]
3. Rewrite the equation by adding and subtracting this square inside the equation:
[tex]\[
x^2 + 2x + 1 - 1 - 6 = 0
\][/tex]
[tex]\[
(x + 1)^2 - 7 = 0
\][/tex]
4. Rearrange to isolate the square term:
[tex]\[
(x + 1)^2 = 7
\][/tex]
5. Solve for [tex]\(x\)[/tex] by taking the square root on both sides:
[tex]\[
x + 1 = \pm \sqrt{7}
\][/tex]
- This gives us two solutions:
[tex]\[
x + 1 = \sqrt{7} \quad \text{or} \quad x + 1 = -\sqrt{7}
\][/tex]
6. Isolate [tex]\(x\)[/tex] by subtracting 1 from both sides:
[tex]\[
x = -1 + \sqrt{7} \quad \text{or} \quad x = -1 - \sqrt{7}
\][/tex]
Therefore, the solutions to the equation are:
(B) [tex]\(x = -1 \pm \sqrt{7}\)[/tex]
These steps demonstrate how to solve the quadratic equation by completing the square, leading to the solution [tex]\(x = -1 \pm \sqrt{7}\)[/tex].
