To solve the equation [tex]\( 0 = 4x^2 - 72x \)[/tex] by completing the square, follow these detailed steps:
1. Rewrite the equation in standard form:
[tex]\[
4x^2 - 72x = 0
\][/tex]
2. Factor out the common factor (if needed):
[tex]\[
4(x^2 - 18x) = 0
\][/tex]
This simplifies the quadratic term.
3. Complete the square inside the parenthesis:
- To complete the square for the expression [tex]\( x^2 - 18x \)[/tex], we need to find the term to add and subtract to make it a perfect square trinomial.
- Take half of the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-18\)[/tex]), and square it:
[tex]\[
\left( \frac{-18}{2} \right)^2 = 81
\][/tex]
- Add and subtract [tex]\( 81 \)[/tex] inside the parenthesis:
[tex]\[
x^2 - 18x + 81 - 81 = 0
\][/tex]
which simplifies to:
[tex]\[
(x - 9)^2 - 81 = 0
\][/tex]
4. Move the constant term to the other side:
[tex]\[
4((x - 9)^2 - 81) = 0
\][/tex]
Simplifying inside the parenthesis yields:
[tex]\[
4(x - 9)^2 - 324 = 0
\][/tex]
Divide both sides by [tex]\( 4 \)[/tex] to isolate the perfect square:
[tex]\[
(x - 9)^2 = 81
\][/tex]
5. Take the square root of both sides:
[tex]\[
x - 9 = \pm 9
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
- For the positive square root:
[tex]\[
x - 9 = 9 \implies x = 18
\][/tex]
- For the negative square root:
[tex]\[
x - 9 = -9 \implies x = 0
\][/tex]
Therefore, the solutions to the equation are [tex]\( x = 0 \)[/tex] and [tex]\( x = 18 \)[/tex].
The correct answer is:
B. [tex]\( x = 0, 18 \)[/tex]
