To complete the square for the equation \( y = x^2 - 16x + 14 \), we follow these steps:
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Start with the quadratic expression: \( x^2 - 16x \).
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Take half of the coefficient of \( x \) (which is -16), square it, and add and subtract this value inside the equation.
Half of -16 is -8, and squaring it gives: \[ (-8)^2 = 64 \]
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Rewrite the quadratic expression by adding and subtracting 64: \[ y = (x^2 - 16x + 64) + 14 - 64 \]
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Now, this can be factored: \[ y = (x - 8)^2 - 50 \]
Now we have the equation in vertex form \( y = (x - h)^2 + k \), where \( (h, k) \) is the vertex.
From \( y = (x - 8)^2 - 50 \):
- \( h = 8 \)
- \( k = -50 \)
Thus, the vertex of the quadratic function \( y = x^2 - 16x + 14 \) is at the point \((8, -50)\).