Solve 5/2+5/x-2=6/x^2-4

$$\frac { 5 } { 2 } + \frac { 5 } { x - 2 } = \frac { 6 } { x ^ { 2 } - 4 }$$

Solve for x

$$x=\frac{\sqrt{85}}{5}-1\approx 0.843908891$$

$$x=-\frac{\sqrt{85}}{5}-1\approx -2.843908891$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-2\right)\left(x+2\right), the least common multiple of 2,x-2,x^{2}-4.

$$2\left(x-2\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2 by x-2.

$$\left(2x-4\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2x-4 by x+2 and combine like terms.

$$\left(2x^{2}-8\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2x^{2}-8 by \frac{5}{2}.

$$5x^{2}-20+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2x+4 by 5.

$$5x^{2}-20+10x+20=2\times 6$$

Add -20 and 20 to get 0.

$$5x^{2}+10x=2\times 6$$

Multiply 2 and 6 to get 12.

$$5x^{2}+10x=12$$

Subtract 12 from both sides.

$$5x^{2}+10x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 10 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-12\right)}}{2\times 5}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-12\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-10±\sqrt{100-20\left(-12\right)}}{2\times 5}$$

Multiply -20 times -12.

$$x=\frac{-10±\sqrt{100+240}}{2\times 5}$$

Add 100 to 240.

$$x=\frac{-10±\sqrt{340}}{2\times 5}$$

Take the square root of 340.

$$x=\frac{-10±2\sqrt{85}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-10±2\sqrt{85}}{10}$$

Now solve the equation x=\frac{-10±2\sqrt{85}}{10} when ± is plus. Add -10 to 2\sqrt{85}.

$$x=\frac{2\sqrt{85}-10}{10}$$

Divide -10+2\sqrt{85} by 10.

$$x=\frac{\sqrt{85}}{5}-1$$

Now solve the equation x=\frac{-10±2\sqrt{85}}{10} when ± is minus. Subtract 2\sqrt{85} from -10.

$$x=\frac{-2\sqrt{85}-10}{10}$$

Divide -10-2\sqrt{85} by 10.

$$x=-\frac{\sqrt{85}}{5}-1$$

The equation is now solved.

$$x=\frac{\sqrt{85}}{5}-1 x=-\frac{\sqrt{85}}{5}-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-2\right)\left(x+2\right), the least common multiple of 2,x-2,x^{2}-4.

$$2\left(x-2\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2 by x-2.

$$\left(2x-4\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2x-4 by x+2 and combine like terms.

$$\left(2x^{2}-8\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2x^{2}-8 by \frac{5}{2}.

$$5x^{2}-20+\left(2x+4\right)\times 5=2\times 6$$

Use the distributive property to multiply 2x+4 by 5.

$$5x^{2}-20+10x+20=2\times 6$$

Add -20 and 20 to get 0.

$$5x^{2}+10x=2\times 6$$

Multiply 2 and 6 to get 12.

$$5x^{2}+10x=12$$

Divide both sides by 5.

$$\frac{5x^{2}+10x}{5}=\frac{12}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{10}{5}x=\frac{12}{5}$$

Divide 10 by 5.

$$x^{2}+2x=\frac{12}{5}$$

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.

$$x^{2}+2x+1^{2}=\frac{12}{5}+1^{2}$$

Square 1.

$$x^{2}+2x+1=\frac{12}{5}+1$$

Add \frac{12}{5} to 1.

$$x^{2}+2x+1=\frac{17}{5}$$

Factor x^{2}+2x+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}.

$$\left(x+1\right)^{2}=\frac{17}{5}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{17}{5}}$$

Simplify.

$$x+1=\frac{\sqrt{85}}{5} x+1=-\frac{\sqrt{85}}{5}$$

Subtract 1 from both sides of the equation.

$$x=\frac{\sqrt{85}}{5}-1 x=-\frac{\sqrt{85}}{5}-1$$