Question

Rewrite the function by completing the square. [g(x) = x² - x - 6]

Answer 1

Completing the square for the function g(x) = x² - x - 6 involves dividing the x coefficient by 2, squaring it, adding and subtracting this square inside the function, and then simplifying, which results in g(x) = (x - 1/2)² - 25/4.

Step-by-step explanation:

Completing the Square for Quadratic Functions

To rewrite the function g(x) = x² - x - 6 by completing the square, we need to follow these steps:

1. Divide the coefficient of the x term by 2 and square it to find the number that completes the square.
2. Add and subtract this number inside the brackets to maintain the equality of the equation.
3. Rewrite the quadratic expression as the square of a binomial and simplify the constant terms.

First, we divide the coefficient of x, which is -1, by 2, getting -1/2, and square it to get 1/4. Adding and subtracting 1/4 within the function, we have:

g(x) = (x² - x + 1/4) - 1/4 - 6

The quadratic part (x² - x + 1/4) can now be written as (x - 1/2)², so we get:

g(x) = (x - 1/2)² - 1/4 - 6

Simplifying the constant terms, we combine -1/4 and -6 which is equal to -6.25 or -25/4 to get:

g(x) = (x - 1/2)² - 25/4