To express the quadratic expression [tex]\( 2x^2 + 12x + 11 \)[/tex] in the form [tex]\( 2(x + a)^2 + b \)[/tex], we will use the method of completing the square. Here is a detailed step-by-step solution:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[
2x^2 + 12x + 11 = 2(x^2 + 6x) + 11
\][/tex]
2. Complete the square for the expression inside the parentheses:
To complete the square for [tex]\( x^2 + 6x \)[/tex], we need to find a term that makes it a perfect square trinomial. The general approach is to add and subtract the square of half the coefficient of [tex]\( x \)[/tex]. Here, the coefficient of [tex]\( x \)[/tex] is 6, so half of this is 3, and its square is [tex]\( 3^2 = 9 \)[/tex].
3. Rewrite the quadratic expression inside the parentheses as a perfect square trinomial:
[tex]\[
x^2 + 6x + 9 - 9 = (x + 3)^2 - 9
\][/tex]
4. Substitute this back into the original expression:
[tex]\[
2(x^2 + 6x + 9 - 9) + 11 = 2((x + 3)^2 - 9) + 11
\][/tex]
5. Distribute the 2 and simplify the constants:
[tex]\[
2(x + 3)^2 - 2 \cdot 9 + 11 = 2(x + 3)^2 - 18 + 11 = 2(x + 3)^2 - 7
\][/tex]
Thus, the expression [tex]\( 2x^2 + 12x + 11 \)[/tex] can be written in the form [tex]\( 2(x + a)^2 + b \)[/tex] as follows:
[tex]\[
2(x + 3)^2 - 18
\][/tex]
So, the constants are:
[tex]\[
a = 3 \quad \text{and} \quad b = -18.
\][/tex]
