To solve the quadratic equation [tex]\(2x^2 + 12x = 66\)[/tex] by completing the square, follow these step-by-step instructions:
1. Move the constant term to the left side:
[tex]\[
2x^2 + 12x - 66 = 0
\][/tex]
2. Divide the entire equation by 2 to simplify:
[tex]\[
x^2 + 6x - 33 = 0
\][/tex]
3. Move the constant term back to the right side:
[tex]\[
x^2 + 6x = 33
\][/tex]
4. Complete the square:
To complete the square, we need to add [tex]\((\frac{b}{2})^2\)[/tex] to both sides, where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]. Here, [tex]\(b = 6\)[/tex], so [tex]\((\frac{6}{2})^2 = 9\)[/tex].
Adding 9 to both sides:
[tex]\[
x^2 + 6x + 9 = 33 + 9
\][/tex]
[tex]\[
(x + 3)^2 = 42
\][/tex]
5. Take the square root of both sides:
[tex]\[
x + 3 = \pm\sqrt{42}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -3 \pm \sqrt{42}
\][/tex]
Thus, the solutions to the quadratic equation are:
[tex]\[
\begin{aligned}
x &= -3 - \sqrt{42}, \\
x &= -3 + \sqrt{42}.
\end{aligned}
\][/tex]
Given the solutions are in the form [tex]\(x = a - \sqrt{b}\)[/tex] and [tex]\(x = a + \sqrt{b}\)[/tex],
we have:
[tex]\[
\begin{aligned}
a &= -3, \\
b &= 42.
\end{aligned}
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = -3, \quad b = 42.
\][/tex]
