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Rewrite the function by completing the square.

\[g(x)= 4 x^{2} -16 x +7\]
\[g(x)=\]
\[(x+\]
\[)^2+\]

1 answer

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