Which of the following are solutions to the equation below? Check all that apply.

[tex](2x + 3)^2 = 10[/tex]

A. [tex]x = -\frac{\sqrt{7}}{2}[/tex]

B. [tex]x = \sqrt{10} + \frac{3}{2}[/tex]

C. [tex]x = \frac{-\sqrt{10} - 3}{2}[/tex]

D. [tex]x = -\sqrt{10} + \frac{3}{2}[/tex]

E. [tex]x = \frac{\sqrt{7}}{2}[/tex]

F. [tex]x = \frac{\sqrt{10} - 3}{2}[/tex]



Answer :

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.