Question
Solve the following quadratic equation using the perfect square trinomial pattern: 64x2−80x+25=0
.(1 point)
Responses
x=0.625
x equals 0.625
x=−0.625
x equals negative 0.625
x=25
x equals 25
x=−25
.(1 point)
Responses
x=0.625
x equals 0.625
x=−0.625
x equals negative 0.625
x=25
x equals 25
x=−25
Answers
GPT-4o mini
To solve the quadratic equation \(64x^2 - 80x + 25 = 0\) using the perfect square trinomial pattern, we first observe that the equation might be factored as a perfect square.
The general form of a perfect square trinomial is:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
In this case, let's rewrite the given equation:
\[
64x^2 - 80x + 25
\]
This can be rearranged as:
\[
(8x)^2 - 2(8x)(5) + (5)^2
\]
This indicates it can be factored as:
\[
(8x - 5)^2
\]
Now, we rewrite the initial equation:
\[
(8x - 5)^2 = 0
\]
Next, we take the square root of both sides:
\[
8x - 5 = 0
\]
Now we can solve for \(x\):
\[
8x = 5 \\
x = \frac{5}{8} \\
x = 0.625
\]
Thus, the solution to the quadratic equation \(64x^2 - 80x + 25 = 0\) is:
**x equals 0.625**.
The general form of a perfect square trinomial is:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
In this case, let's rewrite the given equation:
\[
64x^2 - 80x + 25
\]
This can be rearranged as:
\[
(8x)^2 - 2(8x)(5) + (5)^2
\]
This indicates it can be factored as:
\[
(8x - 5)^2
\]
Now, we rewrite the initial equation:
\[
(8x - 5)^2 = 0
\]
Next, we take the square root of both sides:
\[
8x - 5 = 0
\]
Now we can solve for \(x\):
\[
8x = 5 \\
x = \frac{5}{8} \\
x = 0.625
\]
Thus, the solution to the quadratic equation \(64x^2 - 80x + 25 = 0\) is:
**x equals 0.625**.