EXERCISE 15 olve for \( x \) : \( \begin{array}{l}\sqrt{x+2}=4 \\ 3 \sqrt{x}=2 \sqrt{3} \\ \sqrt{x^{2}+7}-4=0 \\ \sqrt{x-2}=4-x \\ \sqrt{3 x+13}-1=x \\ \sqrt{4-3 x}=4-3 x\end{array} \)

Solve the equation \( \sqrt{x+2}=4 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{x+2}=4\) - step1: Find the domain: \(\sqrt{x+2}=4,x\geq -2\) - step2: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{x+2}\right)^{2}=4^{2}\) - step3: Evaluate the power: \(x+2=16\) - step4: Move the constant to the right side: \(x=16-2\) - step5: Subtract the numbers: \(x=14\) - step6: Check if the solution is in the defined range: \(x=14,x\geq -2\) - step7: Find the intersection: \(x=14\) - step8: Check the solution: \(x=14\) Solve the equation \( 3 \sqrt{x}=2 \sqrt{3} \). Solve the equation by following steps: - step0: Solve for \(x\): \(3\sqrt{x}=2\sqrt{3}\) - step1: Find the domain: \(3\sqrt{x}=2\sqrt{3},x\geq 0\) - step2: Divide both sides: \(\frac{3\sqrt{x}}{3}=\frac{2\sqrt{3}}{3}\) - step3: Divide the numbers: \(\sqrt{x}=\frac{2\sqrt{3}}{3}\) - step4: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{x}\right)^{2}=\left(\frac{2\sqrt{3}}{3}\right)^{2}\) - step5: Evaluate the power: \(x=\frac{4}{3}\) - step6: Check if the solution is in the defined range: \(x=\frac{4}{3},x\geq 0\) - step7: Find the intersection: \(x=\frac{4}{3}\) - step8: Check the solution: \(x=\frac{4}{3}\) Solve the equation \( \sqrt{x^{2}+7}-4=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{x^{2}+7}-4=0\) - step1: Move the constant to the right side: \(\sqrt{x^{2}+7}=0+4\) - step2: Remove 0: \(\sqrt{x^{2}+7}=4\) - step3: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{x^{2}+7}\right)^{2}=4^{2}\) - step4: Evaluate the power: \(x^{2}+7=16\) - step5: Move the constant to the right side: \(x^{2}=16-7\) - step6: Subtract the numbers: \(x^{2}=9\) - step7: Simplify the expression: \(x=\pm \sqrt{9}\) - step8: Simplify: \(x=\pm 3\) - step9: Separate into possible cases: \(\begin{align}&x=3\\&x=-3\end{align}\) - step10: Check the solution: \(\begin{align}&x=3\\&x=-3\end{align}\) - step11: Rewrite: \(x_{1}=-3,x_{2}=3\) Solve the equation \( \sqrt{x-2}=4-x \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{x-2}=4-x\) - step1: Find the domain: \(\sqrt{x-2}=4-x ,x\geq 2\) - step2: Evaluate: \(\sqrt{x-2}=4-x,4-x\geq 0\) - step3: Evaluate: \(\sqrt{x-2}=4-x,x\leq 4\) - step4: Solve the equation: \(\begin{align}&x=6\\&x=3\end{align},x\leq 4\) - step5: Find the intersection: \(x=3\) - step6: Check if the solution is in the defined range: \(x=3,x\geq 2\) - step7: Find the intersection: \(x=3\) - step8: Check the solution: \(x=3\) Solve the equation \( \sqrt{3 x+13}-1=x \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{3x+13}-1=x\) - step1: Find the domain: \(\sqrt{3x+13}-1=x ,x\geq -\frac{13}{3}\) - step2: Move the constant to the right-hand side: \(\sqrt{3x+13}=x+1\) - step3: Evaluate: \(\sqrt{3x+13}=x+1,x+1\geq 0\) - step4: Evaluate: \(\sqrt{3x+13}=x+1,x\geq -1\) - step5: Solve the equation: \(\begin{align}&x=4\\&x=-3\end{align},x\geq -1\) - step6: Find the intersection: \(x=4\) - step7: Check if the solution is in the defined range: \(x=4,x\geq -\frac{13}{3}\) - step8: Find the intersection: \(x=4\) - step9: Check the solution: \(x=4\) Solve the equation \( \sqrt{4-3 x}=4-3 x \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{4-3x}=4-3x\) - step1: Find the domain: \(\sqrt{4-3x}=4-3x ,x\leq \frac{4}{3}\) - step2: Evaluate: \(\sqrt{4-3x}=4-3x,4-3x\geq 0\) - step3: Evaluate: \(\sqrt{4-3x}=4-3x,x\leq \frac{4}{3}\) - step4: Solve the equation: \(\begin{align}&x=1\\&x=\frac{4}{3}\end{align},x\leq \frac{4}{3}\) - step5: Find the intersection: \(\begin{align}&x=1\\&x=\frac{4}{3}\end{align}\) - step6: Check the solution: \(\begin{align}&x=1\\&x=\frac{4}{3}\end{align}\) - step7: Rewrite: \(x_{1}=1,x_{2}=\frac{4}{3}\) The solutions to the given equations are: 1. \( \sqrt{x+2}=4 \) has the solution \( x=14 \). 2. \( 3 \sqrt{x}=2 \sqrt{3} \) has the solutions \( x=\frac{4}{3} \) or \( x=1.3 \). 3. \( \sqrt{x^{2}+7}-4=0 \) has the solutions \( x=-3 \) or \( x=3 \). 4. \( \sqrt{x-2}=4-x \) has the solution \( x=3 \). 5. \( \sqrt{3 x+13}-1=x \) has the solution \( x=4 \). 6. \( \sqrt{4-3 x}=4-3 x \) has the solutions \( x=1 \) or \( x=\frac{4}{3} \).