Solve 3x+1/x-2=2x-12/x+1

$$\frac { 3 x + 1 } { x - 2 } = \frac { 2 x - 12 } { x + 1 }$$

Solve for x

$$x=\sqrt{123}-10\approx 1.090536506$$

$$x=-\left(\sqrt{123}+10\right)\approx -21.090536506$$

Steps Using the Quadratic Formula

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

$$\left(x+1\right)\left(3x+1\right)=\left(x-2\right)\left(2x-12\right)$$

Use the distributive property to multiply x+1 by 3x+1 and combine like terms.

$$3x^{2}+4x+1=\left(x-2\right)\left(2x-12\right)$$

Use the distributive property to multiply x-2 by 2x-12 and combine like terms.

$$3x^{2}+4x+1=2x^{2}-16x+24$$

Subtract 2x^{2} from both sides.

$$3x^{2}+4x+1-2x^{2}=-16x+24$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}+4x+1=-16x+24$$

Add 16x to both sides.

$$x^{2}+4x+1+16x=24$$

Combine 4x and 16x to get 20x.

$$x^{2}+20x+1=24$$

Subtract 24 from both sides.

$$x^{2}+20x+1-24=0$$

Subtract 24 from 1 to get -23.

$$x^{2}+20x-23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-23\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-23\right)}}{2}$$

Multiply -4 times -23.

$$x=\frac{-20±\sqrt{400+92}}{2}$$

Add 400 to 92.

$$x=\frac{-20±\sqrt{492}}{2}$$

Take the square root of 492.

$$x=\frac{-20±2\sqrt{123}}{2}$$

Now solve the equation x=\frac{-20±2\sqrt{123}}{2} when ± is plus. Add -20 to 2\sqrt{123}.

$$x=\frac{2\sqrt{123}-20}{2}$$

Divide -20+2\sqrt{123} by 2.

$$x=\sqrt{123}-10$$

Now solve the equation x=\frac{-20±2\sqrt{123}}{2} when ± is minus. Subtract 2\sqrt{123} from -20.

$$x=\frac{-2\sqrt{123}-20}{2}$$

Divide -20-2\sqrt{123} by 2.

$$x=-\sqrt{123}-10$$

The equation is now solved.

$$x=\sqrt{123}-10 x=-\sqrt{123}-10$$

Steps for Completing the Square

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

$$\left(x+1\right)\left(3x+1\right)=\left(x-2\right)\left(2x-12\right)$$

Use the distributive property to multiply x+1 by 3x+1 and combine like terms.

$$3x^{2}+4x+1=\left(x-2\right)\left(2x-12\right)$$

Use the distributive property to multiply x-2 by 2x-12 and combine like terms.

$$3x^{2}+4x+1=2x^{2}-16x+24$$

Subtract 2x^{2} from both sides.

$$3x^{2}+4x+1-2x^{2}=-16x+24$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}+4x+1=-16x+24$$

Add 16x to both sides.

$$x^{2}+4x+1+16x=24$$

Combine 4x and 16x to get 20x.

$$x^{2}+20x+1=24$$

Subtract 1 from both sides.

$$x^{2}+20x=24-1$$

Subtract 1 from 24 to get 23.

$$x^{2}+20x=23$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=23+10^{2}$$

Square 10.

$$x^{2}+20x+100=23+100$$

Add 23 to 100.

$$x^{2}+20x+100=123$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=123$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{123}$$

Simplify.

$$x+10=\sqrt{123} x+10=-\sqrt{123}$$

Subtract 10 from both sides of the equation.

$$x=\sqrt{123}-10 x=-\sqrt{123}-10$$