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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]
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