To determine which equation is equivalent to the given equation [tex]\( x^2 - 6x = -8 \)[/tex], we will complete the square. Follow these steps:
1. Start with the given equation:
[tex]\[
x^2 - 6x = -8
\][/tex]
2. To complete the square, we need to form a perfect square trinomial on the left side. Begin by isolating the quadratic and linear terms:
[tex]\[
x^2 - 6x
\][/tex]
3. Take the coefficient of [tex]\( x \)[/tex], which is [tex]\(-6\)[/tex], divide it by 2, and square the result:
[tex]\[
\left(-\frac{6}{2}\right)^2 = 9
\][/tex]
4. Add and subtract this square (9) inside the equation to keep it balanced:
[tex]\[
x^2 - 6x + 9 - 9 = -8
\][/tex]
5. Rewrite the perfect square trinomial as a square of a binomial:
[tex]\[
(x - 3)^2 - 9 = -8
\][/tex]
6. Move the constant term [tex]\(-9\)[/tex] to the other side of the equation:
[tex]\[
(x - 3)^2 - 9 = -8
\][/tex]
[tex]\[
(x - 3)^2 = -8 + 9
\][/tex]
[tex]\[
(x - 3)^2 = 1
\][/tex]
Now compare this with the provided answer choices. None match. So it looks like there's an error in given options, since correct answer doesn't match any.
