To complete the square for the quadratic equation \( y = x^2 - 16x + 14 \), follow these steps:
-
Identify the necessary components: Start with the \( x^2 \) and \( -16x \) part.
\[ y = x^2 - 16x + 14 \]
-
Complete the square for \( x^2 - 16x \):
- Take the coefficient of \( x \), which is \(-16\), divide it by 2, and then square it: \[ \left(-\frac{16}{2}\right)^2 = (-8)^2 = 64 \]
-
Rewrite the equation:
- Add and subtract 64 inside the equation: \[ y = (x^2 - 16x + 64) - 64 + 14 \]
-
Simplify:
- The expression \( (x^2 - 16x + 64) \) can be factored as \( (x - 8)^2 \): \[ y = (x - 8)^2 - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]
-
Identify the vertex:
- The equation is now in the vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
- Here, \( h = 8 \) and \( k = -50 \).
Thus, the vertex of the parabola represented by the equation \( y = x^2 - 16x + 14 \) is \((8, -50)\).