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.
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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 \( (a - b)^2 = a^2 - 2ab + b^2 \).
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Determine what \( a \) and \( b \) would be:
- From \( 2ab = 80 \), we can solve for \( b \): \[ 2(8x)(5) = 80 \text{, which is correct.} \]
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Rewrite the equation: The left side can be rewritten as a square: \[ (8x - 5)^2 = 0 \]
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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