Question

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

x=25
x equals 25

x=−25
x equals negative 25

x=−0.625
x equals negative 0.625

x=0.625

Answer 1

To solve the quadratic equation \( 64x^2 - 80x + 25 = 0 \) using the perfect square trinomial pattern, we can first rewrite the left side in a way that helps us identify it as a square.

1. **Identify the coefficients**:
- The first term is \( 64x^2 \) which is \( (8x)^2 \).
- The last term is \( 25 \), which is \( 5^2 \).
- The middle term is \( -80x \), and we know the formula for a perfect square trinomial is \( (a - b)^2 = a^2 - 2ab + b^2 \).

2. **Determine what \( a \) and \( b \) would be**:
- From \( 2ab = 80 \), we can solve for \( b \):
\[
2(8x)(5) = 80 \text{, which is correct.}
\]

3. **Rewrite the equation**:
The left side can be rewritten as a square:
\[
(8x - 5)^2 = 0
\]

4. **Solve for \( x \)**:
To solve \( (8x - 5)^2 = 0 \), we take the square root of both sides:
\[
8x - 5 = 0
\]
Then, solving for \( x \):
\[
8x = 5
\]
\[
x = \frac{5}{8} = 0.625
\]

Thus, the solution to the quadratic equation \( 64x^2 - 80x + 25 = 0 \) is:

\[ x = 0.625 \]

So the correct response is:

**x=0.625**