Solve each equation by completing the square. 1) \( p^{2}+14 p-38=0 \) 3) \( a^{2}+14 a-51=0 \) 5) \( x^{2}+6 x+8=0 \) 7) \( x^{2}+14 x-15=0 \) 15

Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x^{2}+14x-15=0\) - step1: Factor the expression: \(\left(x-1\right)\left(x+15\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-1=0\\&x+15=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=1\\&x=-15\end{align}\) - step4: Rewrite: \(x_{1}=-15,x_{2}=1\) Solve the equation \( a^{2}+14 a-51=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(a^{2}+14a-51=0\) - step1: Factor the expression: \(\left(a-3\right)\left(a+17\right)=0\) - step2: Separate into possible cases: \(\begin{align}&a-3=0\\&a+17=0\end{align}\) - step3: Solve the equation: \(\begin{align}&a=3\\&a=-17\end{align}\) - step4: Rewrite: \(a_{1}=-17,a_{2}=3\) Solve the equation \( x^{2}+6 x+8=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x^{2}+6x+8=0\) - step1: Factor the expression: \(\left(x+2\right)\left(x+4\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x+2=0\\&x+4=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=-2\\&x=-4\end{align}\) - step4: Rewrite: \(x_{1}=-4,x_{2}=-2\) Solve the equation \( p^{2}+14 p-38=0 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(p^{2}+14p-38=0\) - step1: Solve using the quadratic formula: \(p=\frac{-14\pm \sqrt{14^{2}-4\left(-38\right)}}{2}\) - step2: Simplify the expression: \(p=\frac{-14\pm \sqrt{348}}{2}\) - step3: Simplify the expression: \(p=\frac{-14\pm 2\sqrt{87}}{2}\) - step4: Separate into possible cases: \(\begin{align}&p=\frac{-14+2\sqrt{87}}{2}\\&p=\frac{-14-2\sqrt{87}}{2}\end{align}\) - step5: Simplify the expression: \(\begin{align}&p=-7+\sqrt{87}\\&p=\frac{-14-2\sqrt{87}}{2}\end{align}\) - step6: Simplify the expression: \(\begin{align}&p=-7+\sqrt{87}\\&p=-7-\sqrt{87}\end{align}\) - step7: Rewrite: \(p_{1}=-7-\sqrt{87},p_{2}=-7+\sqrt{87}\) Here are the solutions for each equation by completing the square: 1) For the equation \( p^{2}+14 p-38=0 \): - Completing the square gives us: \[ p = -7 \pm \sqrt{87} \] - Thus, the solutions are: \[ p_{1} = -7 - \sqrt{87}, \quad p_{2} = -7 + \sqrt{87} \] 3) For the equation \( a^{2}+14 a-51=0 \): - Completing the square gives us: \[ a = -7 \pm \sqrt{100} \] - Thus, the solutions are: \[ a_{1} = -17, \quad a_{2} = 3 \] 5) For the equation \( x^{2}+6 x+8=0 \): - Completing the square gives us: \[ x = -3 \pm \sqrt{1} \] - Thus, the solutions are: \[ x_{1} = -4, \quad x_{2} = -2 \] 7) For the equation \( x^{2}+14 x-15=0 \): - Completing the square gives us: \[ x = -7 \pm \sqrt{64} \] - Thus, the solutions are: \[ x_{1} = -15, \quad x_{2} = 1 \] These are the solutions for each of the equations provided.