Solve the following quadratic equation for all values of xx in simplest form.

Question

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: \[ x = 1 \quad \text{and} \quad x = \frac{1}{3} \]