Solve 1/(2+alpha)(-4+alpha)=2

$$\frac { 1 } { ( 2 + \alpha ) ( - 4 + \alpha ) } = 2$$

Solve for α

$$\alpha =\frac{\sqrt{38}}{2}+1\approx 4.082207001$$

$$\alpha =-\frac{\sqrt{38}}{2}+1\approx -2.082207001$$

Steps Using the Quadratic Formula

Variable \alpha cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(\alpha -4\right)\left(\alpha +2\right).

$$1=2\left(\alpha -4\right)\left(\alpha +2\right)$$

Use the distributive property to multiply 2 by \alpha -4.

$$1=\left(2\alpha -8\right)\left(\alpha +2\right)$$

Use the distributive property to multiply 2\alpha -8 by \alpha +2 and combine like terms.

$$1=2\alpha ^{2}-4\alpha -16$$

Swap sides so that all variable terms are on the left hand side.

$$2\alpha ^{2}-4\alpha -16=1$$

Subtract 1 from both sides.

$$2\alpha ^{2}-4\alpha -16-1=0$$

Subtract 1 from -16 to get -17.

$$2\alpha ^{2}-4\alpha -17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\alpha =\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-17\right)}}{2\times 2}$$

Square -4.

$$\alpha =\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-17\right)}}{2\times 2}$$

Multiply -4 times 2.

$$\alpha =\frac{-\left(-4\right)±\sqrt{16-8\left(-17\right)}}{2\times 2}$$

Multiply -8 times -17.

$$\alpha =\frac{-\left(-4\right)±\sqrt{16+136}}{2\times 2}$$

Add 16 to 136.

$$\alpha =\frac{-\left(-4\right)±\sqrt{152}}{2\times 2}$$

Take the square root of 152.

$$\alpha =\frac{-\left(-4\right)±2\sqrt{38}}{2\times 2}$$

The opposite of -4 is 4.

$$\alpha =\frac{4±2\sqrt{38}}{2\times 2}$$

Multiply 2 times 2.

$$\alpha =\frac{4±2\sqrt{38}}{4}$$

Now solve the equation \alpha =\frac{4±2\sqrt{38}}{4} when ± is plus. Add 4 to 2\sqrt{38}.

$$\alpha =\frac{2\sqrt{38}+4}{4}$$

Divide 4+2\sqrt{38} by 4.

$$\alpha =\frac{\sqrt{38}}{2}+1$$

Now solve the equation \alpha =\frac{4±2\sqrt{38}}{4} when ± is minus. Subtract 2\sqrt{38} from 4.

$$\alpha =\frac{4-2\sqrt{38}}{4}$$

Divide 4-2\sqrt{38} by 4.

$$\alpha =-\frac{\sqrt{38}}{2}+1$$

The equation is now solved.

$$\alpha =\frac{\sqrt{38}}{2}+1 \alpha =-\frac{\sqrt{38}}{2}+1$$

Steps for Completing the Square

Variable \alpha cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(\alpha -4\right)\left(\alpha +2\right).

$$1=2\left(\alpha -4\right)\left(\alpha +2\right)$$

Use the distributive property to multiply 2 by \alpha -4.

$$1=\left(2\alpha -8\right)\left(\alpha +2\right)$$

Use the distributive property to multiply 2\alpha -8 by \alpha +2 and combine like terms.

$$1=2\alpha ^{2}-4\alpha -16$$

Swap sides so that all variable terms are on the left hand side.

$$2\alpha ^{2}-4\alpha -16=1$$

Add 16 to both sides.

$$2\alpha ^{2}-4\alpha =1+16$$

Add 1 and 16 to get 17.

$$2\alpha ^{2}-4\alpha =17$$

Divide both sides by 2.

$$\frac{2\alpha ^{2}-4\alpha }{2}=\frac{17}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$\alpha ^{2}+\left(-\frac{4}{2}\right)\alpha =\frac{17}{2}$$

Divide -4 by 2.

$$\alpha ^{2}-2\alpha =\frac{17}{2}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\alpha ^{2}-2\alpha +1=\frac{17}{2}+1$$

Add \frac{17}{2} to 1.

$$\alpha ^{2}-2\alpha +1=\frac{19}{2}$$

Factor \alpha ^{2}-2\alpha +1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\alpha -1\right)^{2}=\frac{19}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\alpha -1\right)^{2}}=\sqrt{\frac{19}{2}}$$

Simplify.

$$\alpha -1=\frac{\sqrt{38}}{2} \alpha -1=-\frac{\sqrt{38}}{2}$$

Add 1 to both sides of the equation.

$$\alpha =\frac{\sqrt{38}}{2}+1 \alpha =-\frac{\sqrt{38}}{2}+1$$