To find an equivalent equation to [tex]\( x^2 - 6x = 8 \)[/tex], we will complete the square. Here is a step-by-step solution for this process:
1. Start with the given equation:
[tex]\[
x^2 - 6x = 8
\][/tex]
2. Complete the square on the left-hand side of the equation:
- Move the constant term (8) to the other side temporarily:
[tex]\[
x^2 - 6x - 8 = 0
\][/tex]
- Focus on the quadratic and linear terms ([tex]\( x^2 - 6x \)[/tex]). We need to express it as a perfect square trinomial.
3. To complete the square for [tex]\( x^2 - 6x \)[/tex]:
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\(-6\)[/tex], divide it by 2, and then square it:
[tex]\[
\left(-\frac{6}{2}\right)^2 = (-3)^2 = 9
\][/tex]
- Add and subtract this squared number inside the equation. This doesn't change the equation because we are adding 9 and subtracting 9 (adding zero in effect):
[tex]\[
x^2 - 6x + 9 - 9 = 8
\][/tex]
- The equation is now:
[tex]\[
(x - 3)^2 - 9 = 8
\][/tex]
4. Simplify the equation by moving the constant term (9) to the other side:
[tex]\[
(x - 3)^2 - 9 = 8
\][/tex]
[tex]\[
(x - 3)^2 = 8 + 9
\][/tex]
[tex]\[
(x - 3)^2 = 17
\][/tex]
Therefore, the correct equivalent equation is
[tex]\[
(x - 3)^2 = 17.
\][/tex]
The answer is:
D. [tex]\((x - 3)^2 = 17\)[/tex]
