Question

Solve the following quadratic equation for all values of xx in simplest form.
5, left parenthesis, 3, x, minus, 2, right parenthesis, squared, plus, 27, equals, 32
5(3x−2)
2
+27=
32

Answer 1

To solve the quadratic equation \( 5(3x - 2)^2 + 27 = 32 \), we'll first isolate the quadratic term.

1. Subtract 27 from both sides:
\[
5(3x - 2)^2 = 32 - 27
\]
\[
5(3x - 2)^2 = 5
\]

2. Divide both sides by 5:
\[
(3x - 2)^2 = 1
\]

3. Take the square root of both sides:
\[
3x - 2 = 1 \quad \text{or} \quad 3x - 2 = -1
\]

4. Solve each case separately:

- For \( 3x - 2 = 1 \):
\[
3x = 1 + 2
\]
\[
3x = 3
\]
\[
x = 1
\]

- For \( 3x - 2 = -1 \):
\[
3x = -1 + 2
\]
\[
3x = 1
\]
\[
x = \frac{1}{3}
\]

So the solutions to the quadratic equation \( 5(3x - 2)^2 + 27 = 32 \) are:
\[
x = 1 \quad \text{and} \quad x = \frac{1}{3}
\]