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The vertex form of h(x) = x2 – 14x + 6 is h(x) = (x –

)2 –
.

1 answer

To convert the quadratic function \( h(x) = x^2 - 14x + 6 \) into vertex form \( h(x) = (x - h)^2 + k \), we can complete the square.

  1. Start with the original function: \[ h(x) = x^2 - 14x + 6 \]

  2. To complete the square, take the coefficient of \( x \) (which is \(-14\)), divide it by 2, and square it: \[ \left(-\frac{14}{2}\right)^2 = (-7)^2 = 49 \]

  3. Rewrite the quadratic by adding and subtracting \( 49 \): \[ h(x) = (x^2 - 14x + 49) - 49 + 6 \] \[ h(x) = (x - 7)^2 - 49 + 6 \] \[ h(x) = (x - 7)^2 - 43 \]

Thus, in vertex form, we have: \[ h(x) = (x - 7)^2 - 43 \]

So the values for the blanks are:

  • \( (x - 7)^2 \)
  • \( -43 \)

Final answer: \( h(x) = (x - 7)^2 - 43 \)