To complete the square for the quadratic equation \( y = x^2 - 16x + 14 \), we can follow these steps:
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Start with the given equation: \[ y = x^2 - 16x + 14 \]
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Move the constant \( 14 \) to the other side, but keep it for reference: \[ y - 14 = x^2 - 16x \]
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To complete the square on the right side, we take half of the coefficient of \( x \), which is \( -16 \), divide it by 2 to get \( -8 \), and then square it to get \( 64 \).
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Add and subtract \( 64 \) inside the equation: \[ y - 14 = (x^2 - 16x + 64) - 64 \] \[ y - 14 = (x - 8)^2 - 64 \]
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Now, simplify: \[ y = (x - 8)^2 - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]
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The equation is now in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Thus, the vertex is: \[ (8, -50) \]
Therefore, the vertex of the parabola is \(\boxed{(8, -50)}\).