To solve this problem, we need to understand how to identify the correct quadratic equation from the given quadratic formula application.
The quadratic formula used to find the roots of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In this specific problem, the quadratic formula provided is:
[tex]\[
x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 3 \cdot (-2)}}{2 \cdot 3}
\][/tex]
First, we identify the coefficients:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 8\)[/tex]
- [tex]\(c = -2\)[/tex]
We use these coefficients in our standard form of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
3x^2 + 8x - 2 = 0
\][/tex]
Now, comparing it with the options provided:
A. [tex]\(-2x^2 - 8 = 10x - 3\)[/tex]
Simplifying A:
[tex]\[
-2x^2 - 8 = 10x - 3 \implies -2x^2 - 10x + 3 - 8 = 0 \implies -2x^2 - 10x - 5 = 0
\][/tex]
The coefficients here are different from those in the provided quadratic formula. So, this is not the correct option.
B. [tex]\(3x^2 - 8x -10 = 4\)[/tex]
Simplifying B:
[tex]\[
3x^2 - 8x - 10 = 4 \implies 3x^2 - 8x - 10 - 4 = 0 \implies 3x^2 - 8x - 14 = 0
\][/tex]
The coefficients again do not match those in the provided quadratic formula. Hence, this is not the correct option either.
C. [tex]\(3x^2 + 8x - 10 = -8\)[/tex]
Simplifying C:
[tex]\[
3x^2 + 8x - 10 = -8 \implies 3x^2 + 8x - 10 + 8 = 0 \implies 3x^2 + 8x - 2 = 0
\][/tex]
This equation is the same as [tex]\(3x^2 + 8x - 2 = 0\)[/tex], which has consistent coefficients with the quadratic formula provided:
[tex]\(a = 3\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = -2\)[/tex].
This matches perfectly, thus making C the correct option.
D. [tex]\(-2x^2 + 8x - 3 = 4\)[/tex]
Simplifying D:
[tex]\[
-2x^2 + 8x - 3 = 4 \implies -2x^2 + 8x - 3 - 4 = 0 \implies -2x^2 + 8x - 7 = 0
\][/tex]
The coefficients here do not match the ones in the provided quadratic formula. Therefore, this is also not the correct option.
Thus, the correct equation that could be solved using this application of the quadratic formula is:
C. [tex]\(3x^2 + 8x - 10 = -8\)[/tex]
