To solve the equation [tex]\( 0 = 4x^2 - 72x \)[/tex] by completing the square, follow these steps:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex]:
[tex]\[
0 = 4(x^2 - 18x)
\][/tex]
Divide each side of the equation by 4 to simplify:
[tex]\[
0 = x^2 - 18x
\][/tex]
2. Complete the square inside the parentheses:
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\(-18\)[/tex], divide by 2, and square it:
[tex]\[
\left(\frac{-18}{2}\right)^2 = (-9)^2 = 81
\][/tex]
- Add and subtract this value (81) inside the parentheses:
[tex]\[
x^2 - 18x + 81 - 81
\][/tex]
- Rewrite it as a perfect square and the remaining constant:
[tex]\[
0 = (x - 9)^2 - 81
\][/tex]
3. Isolate the perfect square term:
[tex]\[
(x - 9)^2 - 81 = 0
\][/tex]
4. Move the constant term to the other side:
[tex]\[
(x - 9)^2 = 81
\][/tex]
5. Take the square root of both sides:
[tex]\[
x - 9 = \pm \sqrt{81}
\][/tex]
Since [tex]\(\sqrt{81} = 9\)[/tex], we have:
[tex]\[
x - 9 = \pm 9
\][/tex]
6. Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
- For the positive root: [tex]\( x - 9 = 9 \)[/tex]:
[tex]\[
x = 9 + 9 = 18
\][/tex]
- For the negative root: [tex]\( x - 9 = -9 \)[/tex]:
[tex]\[
x = 9 - 9 = 0
\][/tex]
The solutions to the equation are:
[tex]\[
x = 0 \text{ and } x = 18
\][/tex]
Thus, the correct answer is:
A. [tex]\( x = 0, 18 \)[/tex]
