\( 36 x^{2}+7 x=-2 \) 4 \( 7 x^{2}+24 x=15 \) 5. \( (x+2)(x-3) 50 \)

Solve the equation \( 36x^2+7x=-2 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(36x^{2}+7x=-2\) - step1: Move the expression to the left side: \(36x^{2}+7x+2=0\) - step2: Solve using the quadratic formula: \(x=\frac{-7\pm \sqrt{7^{2}-4\times 36\times 2}}{2\times 36}\) - step3: Simplify the expression: \(x=\frac{-7\pm \sqrt{7^{2}-4\times 36\times 2}}{72}\) - step4: Simplify the expression: \(x=\frac{-7\pm \sqrt{-239}}{72}\) - step5: Simplify the expression: \(x=\frac{-7\pm \sqrt{239}\times i}{72}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{-7+\sqrt{239}\times i}{72}\\&x=\frac{-7-\sqrt{239}\times i}{72}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=-\frac{7}{72}+\frac{\sqrt{239}}{72}i\\&x=\frac{-7-\sqrt{239}\times i}{72}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=-\frac{7}{72}+\frac{\sqrt{239}}{72}i\\&x=-\frac{7}{72}-\frac{\sqrt{239}}{72}i\end{align}\) - step9: Rewrite: \(x_{1}=-\frac{7}{72}-\frac{\sqrt{239}}{72}i,x_{2}=-\frac{7}{72}+\frac{\sqrt{239}}{72}i\) - step10: Remove the complex number(s): \(\textrm{No real solution}\) Solve the equation \( 7x^2+24x=15 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(7x^{2}+24x=15\) - step1: Move the expression to the left side: \(7x^{2}+24x-15=0\) - step2: Solve using the quadratic formula: \(x=\frac{-24\pm \sqrt{24^{2}-4\times 7\left(-15\right)}}{2\times 7}\) - step3: Simplify the expression: \(x=\frac{-24\pm \sqrt{24^{2}-4\times 7\left(-15\right)}}{14}\) - step4: Simplify the expression: \(x=\frac{-24\pm \sqrt{996}}{14}\) - step5: Simplify the expression: \(x=\frac{-24\pm 2\sqrt{249}}{14}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{-24+2\sqrt{249}}{14}\\&x=\frac{-24-2\sqrt{249}}{14}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=\frac{-12+\sqrt{249}}{7}\\&x=\frac{-24-2\sqrt{249}}{14}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=\frac{-12+\sqrt{249}}{7}\\&x=-\frac{12+\sqrt{249}}{7}\end{align}\) - step9: Rewrite: \(x_{1}=-\frac{12+\sqrt{249}}{7},x_{2}=\frac{-12+\sqrt{249}}{7}\) Solve the equation \( (x+2)(x-3) = 50 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(\left(x+2\right)\left(x-3\right)=50\) - step1: Expand the expression: \(x^{2}-x-6=50\) - step2: Move the expression to the left side: \(x^{2}-x-56=0\) - step3: Factor the expression: \(\left(x-8\right)\left(x+7\right)=0\) - step4: Separate into possible cases: \(\begin{align}&x-8=0\\&x+7=0\end{align}\) - step5: Solve the equation: \(\begin{align}&x=8\\&x=-7\end{align}\) - step6: Rewrite: \(x_{1}=-7,x_{2}=8\) The solutions to the given equations are: 1. \(36x^2+7x=-2\) has no real solution. 2. \(7x^2+24x=15\) has two real solutions: \(x_{1}=-\frac{12+\sqrt{249}}{7}\) and \(x_{2}=\frac{-12+\sqrt{249}}{7}\). 3. \((x+2)(x-3) = 50\) has two real solutions: \(x_{1}=-7\) and \(x_{2}=8\).