To solve the quadratic equation \( 64x^2 - 80x + 25 = 0 \) using the perfect square trinomial pattern, we first rewrite the equation by identifying the coefficients of the terms.
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Observe the structure of the equation: \[ 64x^2 - 80x + 25 \]
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Recognize that this can be expressed as a perfect square trinomial: A perfect square trinomial typically has the form \( (ax + b)^2 = a^2x^2 + 2abx + b^2 \). Here, we have:
- \( a^2 = 64 \) (so \( a = 8 \))
- \( b^2 = 25 \) (so \( b = 5 \))
- \( 2ab = -80 \) which checks out because \( 2(8)(-5) = -80 \).
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Rewrite the quadratic: Thus, we can factor the quadratic as: \[ (8x - 5)^2 = 0 \]
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Solve for \( x \): To solve \( (8x - 5)^2 = 0 \), we take the square root of both sides: \[ 8x - 5 = 0 \] Now, solve for \( x \): \[ 8x = 5 \ x = \frac{5}{8} = 0.625 \]
The solution to the equation \( 64x^2 - 80x + 25 = 0 \) is: \[ \boxed{0.625} \]
Therefore, from the provided responses, the correct answer is:
- \( x = 0.625 \)