Certainly! Let's solve the quadratic equation [tex]\( x^2 + 9x + 9 = 0 \)[/tex] step-by-step.
### Step 1: Identify the coefficients
First, identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
For the equation [tex]\( x^2 + 9x + 9 = 0 \)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 9\)[/tex]
- [tex]\(c = 9\)[/tex]
### Step 2: Calculate the discriminant
The discriminant of a quadratic equation is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 9^2 - 4 \times 1 \times 9 \][/tex]
[tex]\[ \Delta = 81 - 36 \][/tex]
[tex]\[ \Delta = 45 \][/tex]
### Step 3: Determine the roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\(a = 1\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(\Delta = 45\)[/tex]:
[tex]\[ x_1 = \frac{-9 + \sqrt{45}}{2 \times 1} \][/tex]
[tex]\[ x_2 = \frac{-9 - \sqrt{45}}{2 \times 1} \][/tex]
### Step 4: Simplify the roots
Since [tex]\(\sqrt{45}\)[/tex] can be written as [tex]\(\sqrt{9 \times 5} = 3\sqrt{5}\)[/tex]:
[tex]\[ x_1 = \frac{-9 + 3\sqrt{5}}{2} \][/tex]
[tex]\[ x_2 = \frac{-9 - 3\sqrt{5}}{2} \][/tex]
### Final Answer
Therefore, the solutions to the quadratic equation [tex]\( x^2 + 9x + 9 = 0 \)[/tex] are:
[tex]\[ x_1 = -1.1458980337503153 \][/tex]
[tex]\[ x_2 = -7.854101966249685 \][/tex]
This numerical evaluation confirms it: The correct choice among the given options is
[tex]\[ \frac{-9 \pm 3 \sqrt{5}}{2} \][/tex].
