> \( x^{2}-5 x+6=0 \) \( >x^{2}-7 x=9 \) \( >2(x+1)^{2}-5=0 \) \( >(2 x-1)(x+3)=2 \) \( >7 x^{2}=2-7 x \) \( >3 x-2 x(x+1)=2 \) NOTE CHECK SOLVE THE FOLLOWING THROUGH COMPLETING THE SQUARE

Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x^{2}-5x+6=0\) - step1: Factor the expression: \(\left(x-3\right)\left(x-2\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-3=0\\&x-2=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=3\\&x=2\end{align}\) - step4: Rewrite: \(x_{1}=2,x_{2}=3\) Solve the equation \( 3 x-2 x(x+1)=2 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(3x-2x\left(x+1\right)=2\) - step1: Expand the expression: \(x-2x^{2}=2\) - step2: Move the expression to the left side: \(x-2x^{2}-2=0\) - step3: Rewrite in standard form: \(-2x^{2}+x-2=0\) - step4: Multiply both sides: \(2x^{2}-x+2=0\) - step5: Solve using the quadratic formula: \(x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 2\times 2}}{2\times 2}\) - step6: Simplify the expression: \(x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 2\times 2}}{4}\) - step7: Simplify the expression: \(x=\frac{1\pm \sqrt{-15}}{4}\) - step8: Simplify the expression: \(x=\frac{1\pm \sqrt{15}\times i}{4}\) - step9: Separate into possible cases: \(\begin{align}&x=\frac{1+\sqrt{15}\times i}{4}\\&x=\frac{1-\sqrt{15}\times i}{4}\end{align}\) - step10: Simplify the expression: \(\begin{align}&x=\frac{1}{4}+\frac{\sqrt{15}}{4}i\\&x=\frac{1-\sqrt{15}\times i}{4}\end{align}\) - step11: Simplify the expression: \(\begin{align}&x=\frac{1}{4}+\frac{\sqrt{15}}{4}i\\&x=\frac{1}{4}-\frac{\sqrt{15}}{4}i\end{align}\) - step12: Rewrite: \(x_{1}=\frac{1}{4}-\frac{\sqrt{15}}{4}i,x_{2}=\frac{1}{4}+\frac{\sqrt{15}}{4}i\) - step13: Remove the complex number(s): \(\textrm{No real solution}\) Solve the equation \( 2(x+1)^{2}-5=0 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(2\left(x+1\right)^{2}-5=0\) - step1: Expand the expression: \(2x^{2}+4x-3=0\) - step2: Solve using the quadratic formula: \(x=\frac{-4\pm \sqrt{4^{2}-4\times 2\left(-3\right)}}{2\times 2}\) - step3: Simplify the expression: \(x=\frac{-4\pm \sqrt{4^{2}-4\times 2\left(-3\right)}}{4}\) - step4: Simplify the expression: \(x=\frac{-4\pm \sqrt{40}}{4}\) - step5: Simplify the expression: \(x=\frac{-4\pm 2\sqrt{10}}{4}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{-4+2\sqrt{10}}{4}\\&x=\frac{-4-2\sqrt{10}}{4}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=\frac{-2+\sqrt{10}}{2}\\&x=\frac{-4-2\sqrt{10}}{4}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=\frac{-2+\sqrt{10}}{2}\\&x=-\frac{2+\sqrt{10}}{2}\end{align}\) - step9: Rewrite: \(x_{1}=-\frac{2+\sqrt{10}}{2},x_{2}=\frac{-2+\sqrt{10}}{2}\) Solve the equation \( x^{2}-7 x=9 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(x^{2}-7x=9\) - step1: Move the expression to the left side: \(x^{2}-7x-9=0\) - step2: Solve using the quadratic formula: \(x=\frac{7\pm \sqrt{\left(-7\right)^{2}-4\left(-9\right)}}{2}\) - step3: Simplify the expression: \(x=\frac{7\pm \sqrt{85}}{2}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{7+\sqrt{85}}{2}\\&x=\frac{7-\sqrt{85}}{2}\end{align}\) - step5: Rewrite: \(x_{1}=\frac{7-\sqrt{85}}{2},x_{2}=\frac{7+\sqrt{85}}{2}\) Solve the equation \( (2 x-1)(x+3)=2 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(\left(2x-1\right)\left(x+3\right)=2\) - step1: Expand the expression: \(2x^{2}+5x-3=2\) - step2: Move the expression to the left side: \(2x^{2}+5x-5=0\) - step3: Solve using the quadratic formula: \(x=\frac{-5\pm \sqrt{5^{2}-4\times 2\left(-5\right)}}{2\times 2}\) - step4: Simplify the expression: \(x=\frac{-5\pm \sqrt{5^{2}-4\times 2\left(-5\right)}}{4}\) - step5: Simplify the expression: \(x=\frac{-5\pm \sqrt{65}}{4}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{-5+\sqrt{65}}{4}\\&x=\frac{-5-\sqrt{65}}{4}\end{align}\) - step7: Rewrite the fraction: \(\begin{align}&x=\frac{-5+\sqrt{65}}{4}\\&x=-\frac{5+\sqrt{65}}{4}\end{align}\) - step8: Rewrite: \(x_{1}=-\frac{5+\sqrt{65}}{4},x_{2}=\frac{-5+\sqrt{65}}{4}\) Solve the equation \( 7 x^{2}=2-7 x \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(7x^{2}=2-7x\) - step1: Move the expression to the left side: \(7x^{2}-2+7x=0\) - step2: Rewrite in standard form: \(7x^{2}+7x-2=0\) - step3: Solve using the quadratic formula: \(x=\frac{-7\pm \sqrt{7^{2}-4\times 7\left(-2\right)}}{2\times 7}\) - step4: Simplify the expression: \(x=\frac{-7\pm \sqrt{7^{2}-4\times 7\left(-2\right)}}{14}\) - step5: Simplify the expression: \(x=\frac{-7\pm \sqrt{105}}{14}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{-7+\sqrt{105}}{14}\\&x=\frac{-7-\sqrt{105}}{14}\end{align}\) - step7: Rewrite the fraction: \(\begin{align}&x=\frac{-7+\sqrt{105}}{14}\\&x=-\frac{7+\sqrt{105}}{14}\end{align}\) - step8: Rewrite: \(x_{1}=-\frac{7+\sqrt{105}}{14},x_{2}=\frac{-7+\sqrt{105}}{14}\) Here are the solutions for each equation using the method of completing the square: 1. **Equation:** \( x^{2}-5 x+6=0 \) - Completing the square gives us: \[ (x-2)(x-3)=0 \] - **Solutions:** \( x_{1}=2, x_{2}=3 \) 2. **Equation:** \( x^{2}-7 x=9 \) - Rearranging gives us: \[ x^{2}-7 x - 9=0 \] - Completing the square results in: \[ x = \frac{7 \pm \sqrt{85}}{2} \] - **Solutions:** \( x_{1}=\frac{7-\sqrt{85}}{2}, x_{2}=\frac{7+\sqrt{85}}{2} \) 3. **Equation:** \( 2(x+1)^{2}-5=0 \) - Rearranging gives us: \[ 2(x+1)^{2}=5 \] - Completing the square results in: \[ x = -1 \pm \frac{\sqrt{10}}{2} \] - **Solutions:** \( x_{1}=-\frac{2+\sqrt{10}}{2}, x_{2}=\frac{-2+\sqrt{10}}{2} \) 4. **Equation:** \( (2 x-1)(x+3)=2 \) - Rearranging gives us: \[ 2x^2 + 5x - 5 = 0 \] - Completing the square results in: \[ x = \frac{-5 \pm \sqrt{65}}{4} \] - **Solutions:** \( x_{1}=-\frac{5+\sqrt{65}}{4}, x_{2}=\frac{-5+\sqrt{65}}{4} \) 5. **Equation:** \( 7 x^{2}=2-7 x \) - Rearranging gives us: \[ 7x^2 + 7x - 2 = 0 \] - Completing the square results in: \[ x = \frac{-7 \pm \sqrt{105}}{14} \] - **Solutions:** \( x_{1}=-\frac{7+\sqrt{105}}{14}, x_{2}=\frac{-7+\sqrt{105}}{14} \) 6. **Equation:** \( 3 x-2 x(x+1)=2 \) - Rearranging gives us: \[ -2x^2 + x - 2 = 0 \] - Completing the square shows that there are no real solutions. - **Solutions:** No real solution These solutions provide the values of \( x \) for each equation as required.