Solve the equation below to find the value(s) of \( z \) \[ 3 z-11 \sqrt{z}-2=-12 \] INSTRUCTION: Write your answers as simplified fractions. If there is more than one answer, write your answers with a semi colon ; between them.

First, let's simplify the equation: \[ 3z - 11\sqrt{z} - 2 = -12 \] Adding 12 to both sides gives: \[ 3z - 11\sqrt{z} + 10 = 0 \] Next, we can let \( \sqrt{z} = x \). Then, \( z = x^2 \), and we can rewrite the equation as: \[ 3x^2 - 11x + 10 = 0 \] Now we can solve this quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 3, b = -11, c = 10 \): \[ x = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 3 \cdot 10}}{2 \cdot 3} \] \[ x = \frac{11 \pm \sqrt{121 - 120}}{6} \] \[ x = \frac{11 \pm 1}{6} \] Calculating the two potential values for \( x \): 1. When using \( 11 + 1 \): \[ x = \frac{12}{6} = 2 \] 2. When using \( 11 - 1 \): \[ x = \frac{10}{6} = \frac{5}{3} \] Now, recall that \( \sqrt{z} = x \): 1. If \( x = 2 \), then \( \sqrt{z} = 2 \) implies \( z = 4 \). 2. If \( x = \frac{5}{3} \), then \( \sqrt{z} = \frac{5}{3} \) implies \( z = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \). Thus, the values of \( z \) are \( 4 ; \frac{25}{9} \).