Question

Solve the following quadratic equation using the perfect square trinomial pattern: 64x2−80x+25=0
.(1 point)
Responses

x=−25
x equals negative 25

x=0.625
x equals 0.625

x=25
x equals 25

x=−0.625

Answer 1

To solve the quadratic equation \( 64x^2 - 80x + 25 = 0 \) using the perfect square trinomial pattern, we need to express it in the form \( (ax + b)^2 = 0 \).

1. Start with the equation:
\[
64x^2 - 80x + 25 = 0
\]

2. Notice that the coefficients suggest that we can factor it. First, check if it is a perfect square trinomial. For the expression to be a perfect square, it should be expressed as \( (ax + b)^2 \).

3. Identify \( a \) and \( b \):
- The first term \( 64x^2 \) can be rewritten as \( (8x)^2 \).
- The last term \( 25 \) can be rewritten as \( 5^2 \).
- The middle term should equal \( 2 \times a \times b \). Here, that would be \( 2 \times (8x) \times 5 = 80x \), which holds.

4. Recognize the perfect square trinomial:
\[
(8x - 5)^2 = 0
\]

5. Set the expression inside the parentheses equal to zero:
\[
8x - 5 = 0
\]

6. Solve for \( x \):
\[
8x = 5 \implies x = \frac{5}{8} = 0.625
\]

The solution to the quadratic equation \( 64x^2 - 80x + 25 = 0 \) is:
\[
x = 0.625
\]

The correct response is:
**x = 0.625**