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

1 answer

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