To complete the square and identify the vertex of the equation y = x^2 - 16x + 14, we first need to take half of the coefficient of x and square it.
Half of -16 is -8, and (-8)^2 is 64.
To complete the square, we add 64 to both sides of the equation:
y + 64 = x^2 - 16x + 14 + 64
Simplifying the right side:
y + 64 = x^2 - 16x + 78
Next, we can rewrite the right side as a perfect square trinomial by factoring the quadratic expression:
y + 64 = (x - 8)^2 + 78 - 64
Simplifying further:
y + 64 = (x - 8)^2 + 14
Finally, we subtract 64 from both sides to isolate y:
y + 64 - 64 = (x - 8)^2 + 14 - 64
y = (x - 8)^2 - 50
The completed square form of the equation is y = (x - 8)^2 - 50.
From this form, we can identify that the vertex of the parabola is at (8, -50).
Complete the square to identify the vertex of y = x^2 – 16x + 14.
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