Sure, let's break down the solution step-by-step and identify any errors.
1. Starting Equation:
[tex]\[
9x + 2 = 8x^2 + 6x
\][/tex]
2. Rearrange to Standard Form:
Move all terms to one side to form a quadratic equation in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
-8x^2 + 3x + 2 = 0
\][/tex]
3. Identify Coefficients:
The quadratic equation in standard form is:
[tex]\[
ax^2 + bx + c = 0
\][/tex]
Here, [tex]\( a = -8 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex].
4. Quadratic Formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Compute the Discriminant:
The discriminant of the quadratic equation is:
[tex]\[
b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
b^2 - 4ac = 3^2 - 4(-8)(2) = 9 + 64 = 73
\][/tex]
6. Apply the Quadratic Formula:
Since the discriminant is positive (73), the roots are real and distinct. Substitute the values into the quadratic formula:
[tex]\[
x = \frac{-3 \pm \sqrt{73}}{-16}
\][/tex]
Simplify the expression for the roots:
[tex]\[
x = \frac{-3 + \sqrt{73}}{-16} \quad \text{and} \quad x = \frac{-3 - \sqrt{73}}{-16}
\][/tex]
Which simplifies to:
[tex]\[
x_1 = \frac{3 - \sqrt{73}}{16} \quad \text{and} \quad x_2 = \frac{3 + \sqrt{73}}{16}
\][/tex]
Evaluating these numerically:
- Calculate [tex]\(x_1 \approx -0.3465\)[/tex]
- Calculate [tex]\(x_2 \approx 0.7215\)[/tex]
7. Numerical Results:
The numerical solution for the roots of the quadratic equation is:
[tex]\[
x_1 \approx -0.3465 \quad \text{and} \quad x_2 \approx 0.7215
\][/tex]
### Comparison with Initial Solution:
- The initial solution contained a mistake in handling the discriminant:
[tex]\[
\frac{-3 \pm \sqrt{9 - 64i}}{-16} \rightarrow \frac{3 \pm \sqrt{55i}}{16}
\][/tex]
This interpretation was incorrect since the discriminant, when evaluated properly (as [tex]\(73\)[/tex]), yields real numbers as roots.
### Conclusion:
Given all the steps correctly, the final step confirms that the roots of the equation [tex]\( -8x^2 + 3x + 2 = 0 \)[/tex] are indeed real numbers and approximately:
[tex]\[
x_1 \approx -0.3465 \quad \text{and} \quad x_2 \approx 0.7215
\][/tex]
The initial solution improperly concluded complex roots. Therefore, the correct final numerical results are real and are:
[tex]\[
-0.3465 \quad \text{and} \quad 0.7215
\][/tex]
