To express the quadratic 2x² - 12x + 7 in the form a(x + b)² + c, we follow these steps:
1. Factor out the coefficient of x² (a) from the first two terms:
2x² - 12x = 2(x² - 6x)
2. Complete the square within the parentheses:
x² - 6x + ___
To complete the square, take half of the coefficient of x (-6/2 = -3) and square it (-3)² = 9.
Add and subtract 9 inside the parentheses:
x² - 6x + 9 - 9
3. Rewrite the expression by factoring the perfect square trinomial and adding the constant term:
2(x² - 6x + 9) - 2(9) + 7
4. Simplify the expression:
2(x - 3)² - 18 + 7
5. Combine constants:
2(x - 3)² - 11
Therefore, the quadratic 2x² - 12x + 7 can be expressed in the form 2(x - 3)² - 11, where a = 2, b = 3, and c = -11.
