Question

Rewrite the function by completing the square.
\[g(x)= 4 x^{2} -16 x +7\]
\[g(x)=\]
\[(x+\]
\[)^2+\]

Answer 1

To rewrite the function \( g(x) = 4x^2 - 16x + 7 \) by completing the square, we can follow these steps:

1. Start by factoring out the coefficient of \( x^2 \) (which is 4) from the first two terms:
\[
g(x) = 4(x^2 - 4x) + 7
\]

2. Next, complete the square for the expression inside the parentheses \( x^2 - 4x \). To do this, we take half of the coefficient of \( x \) (which is \( -4 \)), square it, and add and subtract that value:
- Half of \(-4\) is \(-2\) and squaring it gives us \(4\).
\[
x^2 - 4x = (x^2 - 4x + 4 - 4) = (x - 2)^2 - 4
\]

3. Now substitute this back into the equation:
\[
g(x) = 4((x - 2)^2 - 4) + 7
\]

4. Distribute the 4:
\[
g(x) = 4(x - 2)^2 - 16 + 7
\]

5. Finally, combine the constants:
\[
g(x) = 4(x - 2)^2 - 9
\]

Thus, the function in completed square form is:
\[
g(x) = 4(x - 2)^2 - 9
\]

So, in the desired format:

\[
g(x) = 4 \left( (x - 2)^2 \right) - 9
\]

This corresponds to:
\[
g(x) = 4 \left( (x - 2)^2 \right) + (-9)
\]

Final boxed format:
\[
g(x) = 4 \left( x - 2 \right)^2 - 9
\]