To determine which of the given options are solutions to the equation [tex]\(9x^2 - 64 = 0\)[/tex], we need to solve the equation for [tex]\(x\)[/tex] and then check which of the provided options satisfy the equation.
### Step 1: Solve the Equation
We start with the quadratic equation:
[tex]\[ 9x^2 - 64 = 0 \][/tex]
Add 64 to both sides of the equation to isolate the quadratic term:
[tex]\[ 9x^2 = 64 \][/tex]
Divide both sides by 9 to solve for [tex]\(x^2\)[/tex]:
[tex]\[ x^2 = \frac{64}{9} \][/tex]
Take the square root of both sides to solve for [tex]\(x\)[/tex]. Remember to consider both the positive and negative square roots:
[tex]\[ x = \pm \sqrt{\frac{64}{9}} \][/tex]
[tex]\[ x = \pm \frac{\sqrt{64}}{\sqrt{9}} \][/tex]
[tex]\[ x = \pm \frac{8}{3} \][/tex]
Thus, the solutions for the equation [tex]\(9x^2 - 64 = 0\)[/tex] are:
[tex]\[ x = \frac{8}{3} \quad \text{and} \quad x = -\frac{8}{3} \][/tex]
### Step 2: Verify the Given Options
Now we need to check which of the given options match the solutions [tex]\( x = \frac{8}{3} \)[/tex] and [tex]\( x = -\frac{8}{3} \)[/tex].
A. [tex]\(\frac{3}{8}\)[/tex]
[tex]\[ x = \frac{3}{8} \][/tex]
[tex]\[ 9 \left(\frac{3}{8}\right)^2 - 64 = 9 \cdot \frac{9}{64} - 64 = \frac{81}{64} - 64 \][/tex]
This is not equal to 0, so [tex]\(\frac{3}{8}\)[/tex] is not a solution.
B. -8
[tex]\[ x = -8 \][/tex]
[tex]\[ 9(-8)^2 - 64 = 9 \cdot 64 - 64 = 576 - 64 \][/tex]
This is not equal to 0, so [tex]\(-8\)[/tex] is not a solution.
C. [tex]\(-\frac{3}{8}\)[/tex]
[tex]\[ x = -\frac{3}{8} \][/tex]
[tex]\[ 9 \left(-\frac{3}{8}\right)^2 - 64 = 9 \cdot \frac{9}{64} - 64 = \frac{81}{64} - 64 \][/tex]
This is not equal to 0, so [tex]\(-\frac{3}{8}\)[/tex] is not a solution.
D. [tex]\(-\frac{8}{3}\)[/tex]
[tex]\[ x = -\frac{8}{3} \][/tex]
[tex]\[ 9 \left(-\frac{8}{3}\right)^2 - 64 = 9 \cdot \frac{64}{9} - 64 = 64 - 64 = 0 \][/tex]
This is equal to 0, so [tex]\(-\frac{8}{3}\)[/tex] is a solution.
E. 8
[tex]\[ x = 8 \][/tex]
[tex]\[ 9(8)^2 - 64 = 9 \cdot 64 - 64 = 576 - 64 \][/tex]
This is not equal to 0, so 8 is not a solution.
F. [tex]\(\frac{8}{3}\)[/tex]
[tex]\[ x = \frac{8}{3} \][/tex]
[tex]\[ 9 \left(\frac{8}{3}\right)^2 - 64 = 9 \cdot \frac{64}{9} - 64 = 64 - 64 = 0 \][/tex]
This is equal to 0, so [tex]\(\frac{8}{3}\)[/tex] is a solution.
### Solutions
The options that are solutions to the equation [tex]\( 9x^2 - 64 = 0 \)[/tex] are:
- [tex]\( \frac{8}{3} \)[/tex] (Option F)
- [tex]\( -\frac{8}{3} \)[/tex] (Option D)
