To determine the correct solutions to the equation [tex]\((2x + 3)^2 = 10\)[/tex], follow these detailed steps:
1. Start by expanding and simplifying the given equation:
[tex]\[
(2x + 3)^2 = 10
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
2x + 3 = \pm\sqrt{10}
\][/tex]
This gives us two separate equations:
[tex]\[
2x + 3 = \sqrt{10}
\][/tex]
and
[tex]\[
2x + 3 = -\sqrt{10}
\][/tex]
2. Solve the first equation:
[tex]\[
2x + 3 = \sqrt{10}
\][/tex]
Subtract 3 from both sides:
[tex]\[
2x = \sqrt{10} - 3
\][/tex]
Divide by 2:
[tex]\[
x = \frac{\sqrt{10} - 3}{2}
\][/tex]
3. Solve the second equation:
[tex]\[
2x + 3 = -\sqrt{10}
\][/tex]
Subtract 3 from both sides:
[tex]\[
2x = -\sqrt{10} - 3
\][/tex]
Divide by 2:
[tex]\[
x = \frac{-\sqrt{10} - 3}{2}
\][/tex]
4. Now, compare these solutions to the given options:
- [tex]\(C. \quad x = \frac{-\sqrt{10} - 3}{2}\)[/tex] is one of the solutions we found.
- [tex]\(F. \quad x = \frac{\sqrt{10} - 3}{2}\)[/tex] is the other solution we found.
Therefore, the correct solutions to the equation [tex]\((2x + 3)^2 = 10\)[/tex] from the given options are:
- [tex]\(C. \quad x = \frac{-\sqrt{10} - 3}{2}\)[/tex]
- [tex]\(F. \quad x = \frac{\sqrt{10} - 3}{2}\)[/tex]
Thus, the answers are C and F.
