Let's review the solution step-by-step to determine its correctness.
### Step-by-Step Review:
1. Initial Equation:
[tex]\[
9x + 2 = 8x^2 + 6x
\][/tex]
2. Rearrange into Standard Form:
Subtract [tex]\(9x + 2\)[/tex] from both sides:
[tex]\[
0 = 8x^2 + 6x - 9x - 2
\][/tex]
Simplify:
[tex]\[
0 = 8x^2 - 3x - 2
\][/tex]
Rearrange to match standard quadratic form [tex]\(Ax^2 + Bx + C = 0\)[/tex]:
[tex]\[
-8x^2 + 3x + 2 = 0
\][/tex]
3. Identify Coefficients:
For the equation [tex]\(-8x^2 + 3x + 2 = 0\)[/tex], the coefficients are:
[tex]\[
A = -8, \quad B = 3, \quad C = 2
\][/tex]
4. Calculate the Discriminant:
The discriminant [tex]\(D\)[/tex] is given by the formula [tex]\(D = B^2 - 4AC\)[/tex]:
[tex]\[
D = 3^2 - 4(-8)(2)
\][/tex]
Calculate the terms:
[tex]\[
D = 9 - (-64)
\][/tex]
Simplify:
[tex]\[
D = 9 + 64 = 73
\][/tex]
5. Review the Provided Solution:
Let's look at the discriminant step in the provided solution:
[tex]\[
x = \frac{-3 \pm \sqrt{9 - (64)}}{-16}
\][/tex]
This incorrectly sets the discriminant to:
[tex]\[
9 - 64 = -55
\][/tex]
Then it states:
[tex]\[
x = \frac{-3 \pm \sqrt{-55i}}{-16} \text{,}
\][/tex]
which is incorrect because the discriminant calculation steps should be:
[tex]\[
x = \frac{-3 \pm \sqrt{9 + 64}}{-16} = \frac{-3 \pm \sqrt{73}}{-16}
\][/tex]
### Final Roots Calculation:
With the corrected discriminant [tex]\(D = 73\)[/tex]:
[tex]\[
x = \frac{-3 \pm \sqrt{73}}{-16}
\][/tex]
This gives us two roots:
[tex]\[
x_1 = \frac{-3 + \sqrt{73}}{-16} \approx -0.3465
\][/tex]
[tex]\[
x_2 = \frac{-3 - \sqrt{73}}{-16} \approx 0.7215
\][/tex]
### Summary:
The provided solution contains mistakes in both the discriminant calculation and the final solution steps. The correct discriminant is [tex]\(73\)[/tex], and the correct roots of the quadratic equation [tex]\(-8x^2 + 3x + 2 = 0\)[/tex] are approximately [tex]\(-0.3465\)[/tex] and [tex]\(0.7215\)[/tex].
