To determine the solutions of the equation [tex]\(10x^2 + x = 2\)[/tex], we'll take these steps:
1. Rearrange the Equation:
The given equation is [tex]\(10x^2 + x = 2\)[/tex]. To solve for [tex]\(x\)[/tex], we first rearrange it to the standard quadratic form:
[tex]\[
10x^2 + x - 2 = 0
\][/tex]
2. Identify Solutions Using the Quadratic Formula:
For a standard quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], the solutions can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 10\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -2\)[/tex].
3. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is calculated as follows:
[tex]\[
\Delta = b^2 - 4ac = 1^2 - 4(10)(-2) = 1 + 80 = 81
\][/tex]
4. Find the Roots:
Substitute [tex]\(\Delta\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-1 \pm \sqrt{81}}{20} = \frac{-1 \pm 9}{20}
\][/tex]
This provides us with two solutions:
[tex]\[
x = \frac{-1 + 9}{20} = \frac{8}{20} = \frac{2}{5}
\][/tex]
[tex]\[
x = \frac{-1 - 9}{20} = \frac{-10}{20} = \frac{-1}{2}
\][/tex]
5. Compare with Given Choices:
The solutions we found are [tex]\(x = \frac{2}{5}\)[/tex] and [tex]\(x = \frac{-1}{2}\)[/tex].
6. Match with the Answers Provided:
Now, let's match our solutions with the choices given:
- F. [tex]\(\left\{-\frac{1}{2}, \frac{2}{5}\right\}\)[/tex]
- G. [tex]\(\left\{-\frac{2}{5}, \frac{1}{2}\right\}\)[/tex]
- H. [tex]\(\left\{-5, \frac{1}{2}\right\}\)[/tex]
- J. [tex]\(\left\{-\frac{1}{2}, 5\right\}\)[/tex]
The correct set of solutions is [tex]\(\left\{-\frac{1}{2}, \frac{2}{5}\right\}\)[/tex].
So, the correct answer is F.
Select one:
[tex]\[ \boxed{\text{F}} \][/tex]
