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

Question

.(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