Select the correct answer.

Which equation is equivalent to the given equation?

[tex] x^2 - 6x = 8 [/tex]

A. [tex] (x - 3)^2 = 14 [/tex]
B. [tex] (x - 6)^2 = 44 [/tex]
C. [tex] (x - 6)^2 = 20 [/tex]
D. [tex] (x - 3)^2 = 17 [/tex]



Answer :

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]