Solve for \( x: \) \( \begin{array}{ll}1.1 .1 \quad x^{2}+9 x+14=0 \\ 1.1 .2 \quad 4 x^{2}+9 x-3=0 & \text { (correct to TWO decimal places) } \\ 1.1 .3 \quad \sqrt{x^{2}-5}=2 \sqrt{x} \\ \text { Solve for } x \text { and } y \text { in } \\ 3 x-y \text { and } x^{2}+2 x y-y^{2}+2\end{array} \)

Solve the equation \( x^{2}+9x+14=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x^{2}+9x+14=0\) - step1: Factor the expression: \(\left(x+2\right)\left(x+7\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x+2=0\\&x+7=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=-2\\&x=-7\end{align}\) - step4: Rewrite: \(x_{1}=-7,x_{2}=-2\) Solve the equation \( 4x^{2}+9x-3=0 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(4x^{2}+9x-3=0\) - step1: Solve using the quadratic formula: \(x=\frac{-9\pm \sqrt{9^{2}-4\times 4\left(-3\right)}}{2\times 4}\) - step2: Simplify the expression: \(x=\frac{-9\pm \sqrt{9^{2}-4\times 4\left(-3\right)}}{8}\) - step3: Simplify the expression: \(x=\frac{-9\pm \sqrt{129}}{8}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{-9+\sqrt{129}}{8}\\&x=\frac{-9-\sqrt{129}}{8}\end{align}\) - step5: Rewrite the fraction: \(\begin{align}&x=\frac{-9+\sqrt{129}}{8}\\&x=-\frac{9+\sqrt{129}}{8}\end{align}\) - step6: Rewrite: \(x_{1}=-\frac{9+\sqrt{129}}{8},x_{2}=\frac{-9+\sqrt{129}}{8}\) Solve the equation \( \sqrt{x^{2}-5}=2\sqrt{x} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{x^{2}-5}=2\sqrt{x}\) - step1: Find the domain: \(\sqrt{x^{2}-5}=2\sqrt{x},x\geq \sqrt{5}\) - step2: Swap the sides: \(2\sqrt{x}=\sqrt{x^{2}-5}\) - step3: Rewrite the expression: \(\sqrt{x}=\frac{\sqrt{x^{2}-5}}{2}\) - step4: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{x}\right)^{2}=\left(\frac{\sqrt{x^{2}-5}}{2}\right)^{2}\) - step5: Evaluate the power: \(x=\frac{x^{2}-5}{4}\) - step6: Cross multiply: \(x\times 4=x^{2}-5\) - step7: Simplify the equation: \(4x=x^{2}-5\) - step8: Move the expression to the left side: \(4x-\left(x^{2}-5\right)=0\) - step9: Remove the parentheses: \(4x-x^{2}+5=0\) - step10: Factor the expression: \(\left(-x+5\right)\left(x+1\right)=0\) - step11: Separate into possible cases: \(\begin{align}&-x+5=0\\&x+1=0\end{align}\) - step12: Solve the equation: \(\begin{align}&x=5\\&x=-1\end{align}\) - step13: Check if the solution is in the defined range: \(\begin{align}&x=5\\&x=-1\end{align},x\geq \sqrt{5}\) - step14: Find the intersection: \(x=5\) - step15: Check the solution: \(x=5\) Solve the system of equations \( 3x-y=0 \) and \( x^{2}+2xy-y^{2}+2=0 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x-y=0\\x^{2}+2xy-y^{2}+2=0\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}y=3x\\x^{2}+2xy-y^{2}+2=0\end{array}\right.\) - step2: Substitute the value of \(y:\) \(x^{2}+2x\times 3x-\left(3x\right)^{2}+2=0\) - step3: Simplify: \(-2x^{2}+2=0\) - step4: Move the constant to the right side: \(-2x^{2}=0-2\) - step5: Remove 0: \(-2x^{2}=-2\) - step6: Change the signs: \(2x^{2}=2\) - step7: Divide both sides: \(\frac{2x^{2}}{2}=\frac{2}{2}\) - step8: Divide the numbers: \(x^{2}=1\) - step9: Simplify the expression: \(x=\pm \sqrt{1}\) - step10: Simplify: \(x=\pm 1\) - step11: Separate into possible cases: \(x=1\cup x=-1\) - step12: Rearrange the terms: \(\left\{ \begin{array}{l}x=1\\y=3x\end{array}\right.\cup \left\{ \begin{array}{l}x=-1\\y=3x\end{array}\right.\) - step13: Calculate: \(\left\{ \begin{array}{l}x=1\\y=3\end{array}\right.\cup \left\{ \begin{array}{l}x=-1\\y=-3\end{array}\right.\) - step14: Calculate: \(\left\{ \begin{array}{l}x=-1\\y=-3\end{array}\right.\cup \left\{ \begin{array}{l}x=1\\y=3\end{array}\right.\) - step15: Check the solution: \(\left\{ \begin{array}{l}x=-1\\y=-3\end{array}\right.\cup \left\{ \begin{array}{l}x=1\\y=3\end{array}\right.\) - step16: Rewrite: \(\left(x,y\right) = \left(-1,-3\right)\cup \left(x,y\right) = \left(1,3\right)\) The solutions to the given equations are: 1. \( x^{2}+9x+14=0 \) has solutions \( x=-7 \) and \( x=-2 \). 2. \( 4x^{2}+9x-3=0 \) has solutions \( x \approx -2.544727 \) and \( x \approx 0.294727 \). 3. \( \sqrt{x^{2}-5}=2\sqrt{x} \) has solution \( x=5 \). 4. The system of equations \( 3x-y=0 \) and \( x^{2}+2xy-y^{2}+2=0 \) has solutions \( (x,y) = (-1,-3) \) and \( (x,y) = (1,3) \).