To determine which of the given options are solutions to the equation [tex]\(4x^2 - 81 = 0\)[/tex], we need to solve this quadratic equation. Here is a detailed, step-by-step solution:
1. Start with the given equation:
[tex]\[
4x^2 - 81 = 0
\][/tex]
2. Rearrange the equation to isolate the quadratic term:
[tex]\[
4x^2 = 81
\][/tex]
3. Divide both sides of the equation 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]
We found two solutions: [tex]\(x = \frac{9}{2}\)[/tex] and [tex]\(x = -\frac{9}{2}\)[/tex].
Now let's evaluate the provided options to see which ones match these solutions.
A. [tex]\(9\)[/tex] [tex]\(\quad \rightarrow\)[/tex] NOT a solution
B. [tex]\(-\frac{2}{9}\)[/tex] [tex]\(\quad \rightarrow\)[/tex] NOT a solution
C. [tex]\(-\frac{9}{2}\)[/tex] [tex]\(\quad \rightarrow\)[/tex] is a solution
D. [tex]\(\frac{9}{2}\)[/tex] [tex]\(\quad \rightarrow\)[/tex] is a solution
E. [tex]\(\frac{2}{9}\)[/tex] [tex]\(\quad \rightarrow\)[/tex] NOT a solution
F. [tex]\(-9\)[/tex] [tex]\(\quad \rightarrow\)[/tex] NOT a solution
So, the correct solutions from the given options are:
C. [tex]\(-\frac{9}{2}\)[/tex]
D. [tex]\(\frac{9}{2}\)[/tex]
