Let's solve the equation step-by-step to determine which of the given options are solutions to the equation [tex]\(4x^2 - 81 = 0\)[/tex].
1. Start with the equation:
[tex]\[
4x^2 - 81 = 0
\][/tex]
2. Add 81 to both sides to isolate the quadratic term:
[tex]\[
4x^2 = 81
\][/tex]
3. Divide both sides by 4 to solve for [tex]\(x^2\)[/tex]:
[tex]\[
x^2 = \frac{81}{4}
\][/tex]
4. Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{81}{4}}
\][/tex]
5. Simplify the square root:
[tex]\[
x = \pm \frac{\sqrt{81}}{\sqrt{4}}
\][/tex]
[tex]\[
x = \pm \frac{9}{2}
\][/tex]
So, the solutions to the equation [tex]\(4x^2 - 81 = 0\)[/tex] are:
[tex]\[
x = \frac{9}{2} \quad \text{and} \quad x = -\frac{9}{2}
\][/tex]
6. Check the given options to see which ones match our solutions:
- A. [tex]\(\frac{9}{2}\)[/tex] : This is one of the solutions.
- B. [tex]\(\frac{2}{9}\)[/tex] : This is not a solution.
- C. 9 : This is not a solution.
- D. -9 : This is not a solution.
- E. [tex]\(-\frac{2}{9}\)[/tex] : This is not a solution.
- F. [tex]\(-\frac{9}{2}\)[/tex] : This is one of the solutions.
### Therefore, the correct options are:
[tex]\[
\boxed{\frac{9}{2} \text{ (A)} \quad \text{and} \quad -\frac{9}{2} \text{ (F)}}
\][/tex]
