Solve the following quadratic equation using the perfect square trinomial pattern: 64x2−80x+25=0

Question

.(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

Answer 1

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

1. First, notice that we can factor it as a perfect square. The quadratic can be expressed in the form \( (ax + b)^2 \).

2. The given quadratic is \( 64x^2 - 80x + 25 \). We can rewrite this as: \[ (8x - 5)^2 \] To verify: \[ (8x - 5)(8x - 5) = 64x^2 - 40x - 40x + 25 = 64x^2 - 80x + 25 \] So, indeed, \( 64x^2 - 80x + 25 = (8x - 5)^2 \).

3. Set the perfect square trinomial equal to zero: \[ (8x - 5)^2 = 0 \]

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:

\[ \boxed{x = 0.625} \]