Question

Rewrite the function by completing the square.
\[g(x)= x^{2} - x -6\]
\[g(x)=\]
\[(x+\]
\[)^2+\]

Answer 1

To rewrite the function \( g(x) = x^2 - x - 6 \) by completing the square, follow these steps:

1. Start with the expression:
\[
g(x) = x^2 - x - 6
\]

2. Take the coefficient of \( x \) (which is \(-1\)), divide it by 2, and square it:
\[
\left(-\frac{1}{2}\right)^2 = \frac{1}{4}
\]

3. Add and subtract this value inside the function:
\[
g(x) = x^2 - x + \frac{1}{4} - \frac{1}{4} - 6
\]

4. This simplifies to:
\[
g(x) = \left( x - \frac{1}{2} \right)^2 - \frac{1}{4} - 6
\]

5. Combine the constants:
\[
-\frac{1}{4} - 6 = -\frac{1}{4} - \frac{24}{4} = -\frac{25}{4}
\]

6. So, the final expression for \( g(x) \) by completing the square is:
\[
g(x) = \left( x - \frac{1}{2} \right)^2 - \frac{25}{4}
\]

Thus, the completed form is:
\[
g(x) = \left( x - \frac{1}{2} \right)^2 - \frac{25}{4}
\]