Solve x/x+1-2x/1-x=x^2+3/x^2-1+9/4

$$\frac { x } { x + 1 } - \frac { 2 x } { 1 - x } = \frac { x ^ { 2 } + 3 } { x ^ { 2 } - 1 } + \frac { 9 } { 4 }$$

Solve for x

$$x=3$$

Steps Using Factoring By Grouping

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

$$\left(4x-4\right)x-\left(-4-4x\right)\times 2x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply 4x-4 by x.

$$4x^{2}-4x-\left(-4-4x\right)\times 2x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply -4-4x by 2.

$$4x^{2}-4x-\left(-8-8x\right)x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply -8-8x by x.

$$4x^{2}-4x-\left(-8x-8x^{2}\right)=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

To find the opposite of -8x-8x^{2}, find the opposite of each term.

$$4x^{2}-4x+8x+8x^{2}=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Combine -4x and 8x to get 4x.

$$4x^{2}+4x+8x^{2}=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Combine 4x^{2} and 8x^{2} to get 12x^{2}.

$$12x^{2}+4x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply 4 by x^{2}+3.

$$12x^{2}+4x=4x^{2}+12+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Multiply 4 and \frac{9}{4} to get 9.

$$12x^{2}+4x=4x^{2}+12+9\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 9 by x-1.

$$12x^{2}+4x=4x^{2}+12+\left(9x-9\right)\left(x+1\right)$$

Use the distributive property to multiply 9x-9 by x+1 and combine like terms.

$$12x^{2}+4x=4x^{2}+12+9x^{2}-9$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$12x^{2}+4x=13x^{2}+12-9$$

Subtract 9 from 12 to get 3.

$$12x^{2}+4x=13x^{2}+3$$

Subtract 13x^{2} from both sides.

$$12x^{2}+4x-13x^{2}=3$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+4x=3$$

Subtract 3 from both sides.

$$-x^{2}+4x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=4 ab=-\left(-3\right)=3$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=3 b=1$$

Rewrite -x^{2}+4x-3 as \left(-x^{2}+3x\right)+\left(x-3\right).

$$\left(-x^{2}+3x\right)+\left(x-3\right)$$

Factor out -x in -x^{2}+3x.

$$-x\left(x-3\right)+x-3$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(-x+1\right)$$

To find equation solutions, solve x-3=0 and -x+1=0.

$$x=3 x=1$$

Variable x cannot be equal to 1.

$$x=3$$

Steps Using the Quadratic Formula

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

$$\left(4x-4\right)x-\left(-4-4x\right)\times 2x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply 4x-4 by x.

$$4x^{2}-4x-\left(-4-4x\right)\times 2x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply -4-4x by 2.

$$4x^{2}-4x-\left(-8-8x\right)x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply -8-8x by x.

$$4x^{2}-4x-\left(-8x-8x^{2}\right)=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

To find the opposite of -8x-8x^{2}, find the opposite of each term.

$$4x^{2}-4x+8x+8x^{2}=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Combine -4x and 8x to get 4x.

$$4x^{2}+4x+8x^{2}=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Combine 4x^{2} and 8x^{2} to get 12x^{2}.

$$12x^{2}+4x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply 4 by x^{2}+3.

$$12x^{2}+4x=4x^{2}+12+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Multiply 4 and \frac{9}{4} to get 9.

$$12x^{2}+4x=4x^{2}+12+9\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 9 by x-1.

$$12x^{2}+4x=4x^{2}+12+\left(9x-9\right)\left(x+1\right)$$

Use the distributive property to multiply 9x-9 by x+1 and combine like terms.

$$12x^{2}+4x=4x^{2}+12+9x^{2}-9$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$12x^{2}+4x=13x^{2}+12-9$$

Subtract 9 from 12 to get 3.

$$12x^{2}+4x=13x^{2}+3$$

Subtract 13x^{2} from both sides.

$$12x^{2}+4x-13x^{2}=3$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+4x=3$$

Subtract 3 from both sides.

$$-x^{2}+4x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$x=\frac{-4±\sqrt{16-12}}{2\left(-1\right)}$$

Add 16 to -12.

$$x=\frac{-4±\sqrt{4}}{2\left(-1\right)}$$

Take the square root of 4.

$$x=\frac{-4±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±2}{-2}$$

Now solve the equation x=\frac{-4±2}{-2} when ± is plus. Add -4 to 2.

$$x=-\frac{2}{-2}$$

Divide -2 by -2.

$$x=1$$

Now solve the equation x=\frac{-4±2}{-2} when ± is minus. Subtract 2 from -4.

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=1 x=3$$

Variable x cannot be equal to 1.

$$x=3$$

Steps for Completing the Square

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

$$\left(4x-4\right)x-\left(-4-4x\right)\times 2x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply 4x-4 by x.

$$4x^{2}-4x-\left(-4-4x\right)\times 2x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply -4-4x by 2.

$$4x^{2}-4x-\left(-8-8x\right)x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply -8-8x by x.

$$4x^{2}-4x-\left(-8x-8x^{2}\right)=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

To find the opposite of -8x-8x^{2}, find the opposite of each term.

$$4x^{2}-4x+8x+8x^{2}=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Combine -4x and 8x to get 4x.

$$4x^{2}+4x+8x^{2}=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Combine 4x^{2} and 8x^{2} to get 12x^{2}.

$$12x^{2}+4x=4\left(x^{2}+3\right)+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Use the distributive property to multiply 4 by x^{2}+3.

$$12x^{2}+4x=4x^{2}+12+4\left(x-1\right)\left(x+1\right)\times \frac{9}{4}$$

Multiply 4 and \frac{9}{4} to get 9.

$$12x^{2}+4x=4x^{2}+12+9\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 9 by x-1.

$$12x^{2}+4x=4x^{2}+12+\left(9x-9\right)\left(x+1\right)$$

Use the distributive property to multiply 9x-9 by x+1 and combine like terms.

$$12x^{2}+4x=4x^{2}+12+9x^{2}-9$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$12x^{2}+4x=13x^{2}+12-9$$

Subtract 9 from 12 to get 3.

$$12x^{2}+4x=13x^{2}+3$$

Subtract 13x^{2} from both sides.

$$12x^{2}+4x-13x^{2}=3$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+4x=3$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=\frac{3}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=\frac{3}{-1}$$

Divide 3 by -1.

$$x^{2}-4x=-3$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

$$x^{2}-4x+4=-3+4$$

Add -3 to 4.

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

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-2=1 x-2=-1$$

Add 2 to both sides of the equation.

$$x=3 x=1$$

Variable x cannot be equal to 1.

$$x=3$$