To solve for [tex]\(a\)[/tex] given the function [tex]\(h(a) = 9a^2 + 46a\)[/tex] and the condition [tex]\(h(a) = -5\)[/tex], we start by setting up the equation [tex]\(9a^2 + 46a = -5\)[/tex]:
1. Begin by moving [tex]\(-5\)[/tex] to the left side to set the equation to zero:
[tex]\[
9a^2 + 46a + 5 = 0
\][/tex]
2. Now we have a standard quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 9\)[/tex], [tex]\(b = 46\)[/tex], and [tex]\(c = 5\)[/tex].
3. The solutions to the quadratic equation can be found using the quadratic formula:
[tex]\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Substitute [tex]\(a = 9\)[/tex], [tex]\(b = 46\)[/tex], and [tex]\(c = 5\)[/tex] into the quadratic formula:
[tex]\[
a = \frac{-46 \pm \sqrt{46^2 - 4 \cdot 9 \cdot 5}}{18}
\][/tex]
5. Calculate the discriminant:
[tex]\[
46^2 - 4 \cdot 9 \cdot 5 = 2116 - 180 = 1936
\][/tex]
6. Now, take the square root of the discriminant:
[tex]\[
\sqrt{1936} = \sqrt{1936} = 44
\][/tex]
7. Substitute back into the quadratic formula:
[tex]\[
a = \frac{-46 \pm 44}{18}
\][/tex]
8. Solve for the two possible values of [tex]\(a\)[/tex]:
- For the positive root:
[tex]\[
a = \frac{-46 + 44}{18} = \frac{-2}{18} = -\frac{1}{9}
\][/tex]
- For the negative root:
[tex]\[
a = \frac{-46 - 44}{18} = \frac{-90}{18} = -5
\][/tex]
So, the exact solutions to the equation [tex]\(9a^2 + 46a + 5 = 0\)[/tex] are:
[tex]\[
-\frac{1}{9}, -5
\][/tex]
Therefore, the solutions are:
[tex]\[
a = -\frac{23}{9} - \frac{\sqrt{439}}{9}, -\frac{23}{9} + \frac{\sqrt{439}}{9}
\][/tex]
Thus, the exact answers are:
[tex]\[
- \frac{23}{9} - \frac{\sqrt{439}}{9}, - \frac{23}{9} + \frac{\sqrt{439}}{9}
\][/tex]
