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.

Solve the equation \( 3z-11\sqrt{z}-2=-12 \). Solve the equation by following steps: - step0: Solve for \(z\): \(3z-11\sqrt{z}-2=-12\) - step1: Find the domain: \(3z-11\sqrt{z}-2=-12,z\geq 0\) - step2: Move the constant to the right side: \(3z-11\sqrt{z}=-12-\left(-2\right)\) - step3: Calculate: \(3z-11\sqrt{z}=-10\) - step4: Move the expression to the right-hand side: \(-11\sqrt{z}=-10-3z\) - step5: Divide both sides: \(11\sqrt{z}=10+3z\) - step6: Rewrite the expression: \(\sqrt{z}=\frac{10+3z}{11}\) - step7: Evaluate: \(\sqrt{z}=\frac{10+3z}{11},\frac{10+3z}{11}\geq 0\) - step8: Evaluate: \(\sqrt{z}=\frac{10+3z}{11},z\geq -\frac{10}{3}\) - step9: Solve the equation: \(\begin{align}&z=4\\&z=\frac{25}{9}\end{align},z\geq -\frac{10}{3}\) - step10: Find the intersection: \(\begin{align}&z=4\\&z=\frac{25}{9}\end{align}\) - step11: Check if the solution is in the defined range: \(\begin{align}&z=4\\&z=\frac{25}{9}\end{align},z\geq 0\) - step12: Find the intersection: \(\begin{align}&z=4\\&z=\frac{25}{9}\end{align}\) - step13: Check the solution: \(\begin{align}&z=4\\&z=\frac{25}{9}\end{align}\) - step14: Rewrite: \(z_{1}=\frac{25}{9},z_{2}=4\) The solutions to the equation \(3z-11\sqrt{z}-2=-12\) are \(z=\frac{25}{9}\) and \(z=4\).