Question

Solve the following quadratic equation for all values of xx in simplest form.
9, left parenthesis, 5, minus, 3, x, right parenthesis, squared, equals, 16
9(5−3x)
2
=
16

Answer 1

To solve the quadratic equation \( 9(5 - 3x)^2 = 16 \), we will first simplify and solve for \( x \).

1. **Divide both sides by 9**:
\[
(5 - 3x)^2 = \frac{16}{9}
\]

2. **Take the square root of both sides**:
\[
5 - 3x = \pm \frac{4}{3}
\]

This results in two separate equations:

**Case 1**:
\[
5 - 3x = \frac{4}{3}
\]

**Case 2**:
\[
5 - 3x = -\frac{4}{3}
\]

### Solving Case 1:
\[
5 - 3x = \frac{4}{3}
\]

3. **Subtract 5 from both sides**:
\[
-3x = \frac{4}{3} - 5
\]

4. **Convert 5 to a fraction**:
\[
5 = \frac{15}{3}
\]
So,
\[
-3x = \frac{4}{3} - \frac{15}{3} = \frac{4 - 15}{3} = \frac{-11}{3}
\]

5. **Divide both sides by -3**:
\[
x = \frac{-11}{3} \cdot \frac{-1}{3} = \frac{11}{9}
\]

### Solving Case 2:
\[
5 - 3x = -\frac{4}{3}
\]

3. **Subtract 5 from both sides**:
\[
-3x = -\frac{4}{3} - 5
\]

4. **Convert 5 to a fraction**:
\[
-3x = -\frac{4}{3} - \frac{15}{3} = -\frac{4 + 15}{3} = -\frac{19}{3}
\]

5. **Divide both sides by -3**:
\[
x = \frac{19}{3} \cdot \frac{-1}{3} = \frac{19}{9}
\]

### Final Solutions:
The solutions to the quadratic equation \( 9(5 - 3x)^2 = 16 \) are:
\[
x = \frac{11}{9} \quad \text{and} \quad x = \frac{19}{9}
\]