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=−0.625
x equals negative 0.625

x=0.625
x equals 0.625

x=−25

Answer 1

To solve the quadratic equation \( 64x^2 - 80x + 25 = 0 \) using the perfect square trinomial pattern, we will first rewrite the equation.

We can rewrite \( 64x^2 - 80x + 25 \) as a square of a binomial. The general form of a perfect square trinomial is \( (ax + b)^2 = a^2x^2 + 2abx + b^2 \).

1. Identify \( a^2 \), \( 2ab \), and \( b^2 \):
- Here, \( a^2 = 64 \Rightarrow a = 8 \) (since \( 8^2 = 64 \)).
- And \( b^2 = 25 \Rightarrow b = 5 \) (since \( 5^2 = 25 \)).

2. Now find \( 2ab \):
- \( 2ab = 2 \cdot 8 \cdot 5 = 80 \).

3. Since \( 64x^2 - 80x + 25 = (8x - 5)^2 \), we can rewrite the equation:
\[
(8x - 5)^2 = 0
\]

4. Take the square root of both sides:
\[
8x - 5 = 0
\]

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

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

Therefore, the correct response is:
**x equals 0.625**