Question
Solve the equation by completing the square.
0 = 4x2 − 72x
A.
x = -90, 72
B.
x = -72, 90
C.
x = -18, 0
D.
x = 0, 18
0 = 4x2 − 72x
A.
x = -90, 72
B.
x = -72, 90
C.
x = -18, 0
D.
x = 0, 18
Answers
Answered by
GPT-4o mini
To solve the equation \( 0 = 4x^2 - 72x \) by completing the square, we can first rewrite the equation in a more manageable form.
1. **Factor out the 4 from the quadratic expression:**
\[
0 = 4(x^2 - 18x)
\]
2. **Set the factor equal to zero:**
\[
4(x^2 - 18x) = 0
\]
Dividing both sides by 4:
\[
x^2 - 18x = 0
\]
3. **Complete the square:**
To complete the square, we take the coefficient of \(x\), which is -18, halve it to get -9, and then square it to get 81.
\[
x^2 - 18x + 81 = 81
\]
Thus, we rewrite it as:
\[
(x - 9)^2 = 81
\]
4. **Take the square root of both sides:**
\[
x - 9 = \pm 9
\]
This yields two equations:
\[
x - 9 = 9 \quad \Rightarrow \quad x = 18
\]
and
\[
x - 9 = -9 \quad \Rightarrow \quad x = 0
\]
5. **The solutions are:**
\[
x = 0, 18
\]
Thus, the correct answer is \( \text{D. } x = 0, 18 \).
1. **Factor out the 4 from the quadratic expression:**
\[
0 = 4(x^2 - 18x)
\]
2. **Set the factor equal to zero:**
\[
4(x^2 - 18x) = 0
\]
Dividing both sides by 4:
\[
x^2 - 18x = 0
\]
3. **Complete the square:**
To complete the square, we take the coefficient of \(x\), which is -18, halve it to get -9, and then square it to get 81.
\[
x^2 - 18x + 81 = 81
\]
Thus, we rewrite it as:
\[
(x - 9)^2 = 81
\]
4. **Take the square root of both sides:**
\[
x - 9 = \pm 9
\]
This yields two equations:
\[
x - 9 = 9 \quad \Rightarrow \quad x = 18
\]
and
\[
x - 9 = -9 \quad \Rightarrow \quad x = 0
\]
5. **The solutions are:**
\[
x = 0, 18
\]
Thus, the correct answer is \( \text{D. } x = 0, 18 \).