To solve the quadratic equation [tex]\(x^2 + 8x = 20\)[/tex] by completing the square, follow these steps:
1. Start with the given equation:
[tex]\[
x^2 + 8x = 20
\][/tex]
2. Move the constant term to the other side:
[tex]\[
x^2 + 8x = 20
\][/tex]
3. Find the value needed to complete the square.
To complete the square, take half of the coefficient of [tex]\(x\)[/tex], which is 8, divide it by 2 to get 4, and then square it:
[tex]\[
\left( \frac{8}{2} \right)^2 = 4^2 = 16
\][/tex]
4. Add and subtract this value inside the equation:
[tex]\[
x^2 + 8x + 16 - 16 = 20
\][/tex]
5. Rewrite the equation by grouping the perfect square trinomial and keeping the balance:
[tex]\[
x^2 + 8x + 16 = 20 + 16
\][/tex]
[tex]\[
(x + 4)^2 = 36
\][/tex]
6. Take the square root of both sides of the equation:
[tex]\[
x + 4 = \pm \sqrt{36}
\][/tex]
[tex]\[
x + 4 = \pm 6
\][/tex]
7. Solve for [tex]\(x\)[/tex] by isolating the variable:
- For the positive square root:
[tex]\[
x + 4 = 6
\][/tex]
[tex]\[
x = 6 - 4
\][/tex]
[tex]\[
x = 2
\][/tex]
- For the negative square root:
[tex]\[
x + 4 = -6
\][/tex]
[tex]\[
x = -6 - 4
\][/tex]
[tex]\[
x = -10
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 + 8x = 20\)[/tex] are [tex]\(x = 2\)[/tex] and [tex]\(x = -10\)[/tex].
Therefore, the correct answer is:
B. [tex]\(x = 2\)[/tex] or [tex]\(x = -10\)[/tex]
