Solve 1/x-1-1/x+2=5/4

$$\frac{ 1 }{ x-1 } - \frac{ 1 }{ x+2 } = \frac{ 5 }{ 4 }$$

Solve for x

$$x=\frac{\sqrt{465}}{10}-\frac{1}{2}\approx 1.656385865$$

$$x=-\frac{\sqrt{465}}{10}-\frac{1}{2}\approx -2.656385865$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by 4\left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2,4.

$$4x+8-\left(4x-4\right)=5\left(x-1\right)\left(x+2\right)$$

To find the opposite of 4x-4, find the opposite of each term.

$$4x+8-4x+4=5\left(x-1\right)\left(x+2\right)$$

Combine 4x and -4x to get 0.

$$8+4=5\left(x-1\right)\left(x+2\right)$$

Add 8 and 4 to get 12.

$$12=5\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 5 by x-1.

$$12=\left(5x-5\right)\left(x+2\right)$$

Use the distributive property to multiply 5x-5 by x+2 and combine like terms.

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

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

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

Subtract 12 from both sides.

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

Subtract 12 from -10 to get -22.

$$5x^{2}+5x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 5 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 5\left(-22\right)}}{2\times 5}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 5\left(-22\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-5±\sqrt{25-20\left(-22\right)}}{2\times 5}$$

Multiply -20 times -22.

$$x=\frac{-5±\sqrt{25+440}}{2\times 5}$$

Add 25 to 440.

$$x=\frac{-5±\sqrt{465}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-5±\sqrt{465}}{10}$$

Now solve the equation x=\frac{-5±\sqrt{465}}{10} when ± is plus. Add -5 to \sqrt{465}.

$$x=\frac{\sqrt{465}-5}{10}$$

Divide -5+\sqrt{465} by 10.

$$x=\frac{\sqrt{465}}{10}-\frac{1}{2}$$

Now solve the equation x=\frac{-5±\sqrt{465}}{10} when ± is minus. Subtract \sqrt{465} from -5.

$$x=\frac{-\sqrt{465}-5}{10}$$

Divide -5-\sqrt{465} by 10.

$$x=-\frac{\sqrt{465}}{10}-\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{465}}{10}-\frac{1}{2} x=-\frac{\sqrt{465}}{10}-\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by 4\left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2,4.

$$4x+8-\left(4x-4\right)=5\left(x-1\right)\left(x+2\right)$$

To find the opposite of 4x-4, find the opposite of each term.

$$4x+8-4x+4=5\left(x-1\right)\left(x+2\right)$$

Combine 4x and -4x to get 0.

$$8+4=5\left(x-1\right)\left(x+2\right)$$

Add 8 and 4 to get 12.

$$12=5\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 5 by x-1.

$$12=\left(5x-5\right)\left(x+2\right)$$

Use the distributive property to multiply 5x-5 by x+2 and combine like terms.

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

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

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

Add 10 to both sides.

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

Add 12 and 10 to get 22.

$$5x^{2}+5x=22$$

Divide both sides by 5.

$$\frac{5x^{2}+5x}{5}=\frac{22}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{5}{5}x=\frac{22}{5}$$

Divide 5 by 5.

$$x^{2}+x=\frac{22}{5}$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{22}{5}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{22}{5}+\frac{1}{4}$$

Add \frac{22}{5} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{93}{20}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{93}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{93}{20}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{465}}{10} x+\frac{1}{2}=-\frac{\sqrt{465}}{10}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{465}}{10}-\frac{1}{2} x=-\frac{\sqrt{465}}{10}-\frac{1}{2}$$