What is y = x^2 + 12x + 15 in vertex form?
To express the quadratic function y = x^2 + 12x + 15 in vertex form, we need to complete the square by adding and subtracting a constant.
First, we group the x terms and leave space to complete the square:
y = x^2 + 12x + 15
= (x^2 + 12x) + 15
Next, we need to find the constant that when added and subtracted inside the parentheses will complete the square. To determine this constant, we take half of the coefficient of the x-term (12) and square it:
(12/2)^2 = 6^2 = 36
Now we add and subtract 36 inside the parentheses:
= (x^2 + 12x + 36 - 36) + 15
We can rewrite the equation as follows:
= [(x + 6)^2 - 36] + 15
To simplify further, we combine like terms:
= (x + 6)^2 - 36 + 15
= (x + 6)^2 - 21
Therefore, the quadratic function y = x^2 + 12x + 15 in vertex form is y = (x + 6)^2 - 21.