To solve the quadratic equation \( 64x^2 - 80x + 25 = 0 \) using the perfect square trinomial pattern, we need to express it in the form \( (ax + b)^2 = 0 \).
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Start with the equation: \[ 64x^2 - 80x + 25 = 0 \]
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Notice that the coefficients suggest that we can factor it. First, check if it is a perfect square trinomial. For the expression to be a perfect square, it should be expressed as \( (ax + b)^2 \).
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Identify \( a \) and \( b \):
- The first term \( 64x^2 \) can be rewritten as \( (8x)^2 \).
- The last term \( 25 \) can be rewritten as \( 5^2 \).
- The middle term should equal \( 2 \times a \times b \). Here, that would be \( 2 \times (8x) \times 5 = 80x \), which holds.
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Recognize the perfect square trinomial: \[ (8x - 5)^2 = 0 \]
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Set the expression inside the parentheses equal to zero: \[ 8x - 5 = 0 \]
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Solve for \( x \): \[ 8x = 5 \implies x = \frac{5}{8} = 0.625 \]
The solution to the quadratic equation \( 64x^2 - 80x + 25 = 0 \) is: \[ x = 0.625 \]
The correct response is: x = 0.625