Question
Solve the following quadratic equation using the perfect square trinomial pattern: 64x^2−80x+25=0
.(1 point)
Responses
x=−0.625
x equals negative 0.625
x=0.625
x equals 0.625
x=25
x equals 25
x=−25
.(1 point)
Responses
x=−0.625
x equals negative 0.625
x=0.625
x equals 0.625
x=25
x equals 25
x=−25
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \(64x^2 - 80x + 25 = 0\) using the perfect square trinomial pattern, we can rewrite the equation in the form of \((ax + b)^2 = c\).
1. First, observe the quadratic:
\[
64x^2 - 80x + 25.
\]
2. Identify the coefficients:
- \(a^2 = 64\) so \(a = \sqrt{64} = 8\),
- For the linear term, \(-80\), we divide it by \(2a\) (where \(a = 8\)):
\[
\frac{-80}{2 \cdot 8} = \frac{-80}{16} = -5.
\]
3. Rewrite the quadratic using the perfect square trinomial:
\[
(8x - 5)^2 = 0.
\]
4. Setting this equal to 0:
\[
(8x - 5)^2 = 0.
\]
5. Taking the square root of both sides:
\[
8x - 5 = 0.
\]
6. Solving for \(x\):
\[
8x = 5 \\
x = \frac{5}{8}.
\]
Thus, \(x = 0.625\).
### Therefore, the correct response is:
**x = 0.625** or **x equals 0.625**.
1. First, observe the quadratic:
\[
64x^2 - 80x + 25.
\]
2. Identify the coefficients:
- \(a^2 = 64\) so \(a = \sqrt{64} = 8\),
- For the linear term, \(-80\), we divide it by \(2a\) (where \(a = 8\)):
\[
\frac{-80}{2 \cdot 8} = \frac{-80}{16} = -5.
\]
3. Rewrite the quadratic using the perfect square trinomial:
\[
(8x - 5)^2 = 0.
\]
4. Setting this equal to 0:
\[
(8x - 5)^2 = 0.
\]
5. Taking the square root of both sides:
\[
8x - 5 = 0.
\]
6. Solving for \(x\):
\[
8x = 5 \\
x = \frac{5}{8}.
\]
Thus, \(x = 0.625\).
### Therefore, the correct response is:
**x = 0.625** or **x equals 0.625**.
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