Sure, let's solve the equation step by step:
Given equation:
[tex]\[
(w - 4)^2 = 2w^2 + w + 30
\][/tex]
1. First, expand the left-hand side of the equation:
[tex]\[
(w - 4)^2 = w^2 - 8w + 16
\][/tex]
So, our equation becomes:
[tex]\[
w^2 - 8w + 16 = 2w^2 + w + 30
\][/tex]
2. To set the equation to zero, move all the terms to one side. Subtract [tex]\(2w^2 + w + 30\)[/tex] from both sides:
[tex]\[
w^2 - 8w + 16 - 2w^2 - w - 30 = 0
\][/tex]
Combine like terms:
[tex]\[
w^2 - 2w^2 - 8w - w + 16 - 30 = 0
\][/tex]
[tex]\[
-w^2 - 9w - 14 = 0
\][/tex]
3. Multiply the entire equation by -1 to make the leading coefficient positive:
[tex]\[
w^2 + 9w + 14 = 0
\][/tex]
Now, we have a standard quadratic equation:
[tex]\[
w^2 + 9w + 14 = 0
\][/tex]
4. Solve the quadratic equation using the quadratic formula [tex]\((w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = 14\)[/tex].
First, calculate the discriminant:
[tex]\[
b^2 - 4ac = 9^2 - 4 \cdot 1 \cdot 14 = 81 - 56 = 25
\][/tex]
5. Now, apply the quadratic formula:
[tex]\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-9 \pm \sqrt{25}}{2 \cdot 1}
\][/tex]
[tex]\[
w = \frac{-9 \pm 5}{2}
\][/tex]
This gives us two solutions:
[tex]\[
w = \frac{-9 + 5}{2} = \frac{-4}{2} = -2
\][/tex]
[tex]\[
w = \frac{-9 - 5}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions are:
[tex]\[
w = -2 \quad \text{and} \quad w = -7
\][/tex]
