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

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

Solve for x

$$x=\frac{\sqrt{33}-3}{4}\approx 0.686140662$$

$$x=\frac{-\sqrt{33}-3}{4}\approx -2.186140662$$

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 \left(x-1\right)\left(x+1\right), the least common multiple of 1+x,1-x^{2},1-x.

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

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

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

To find the opposite of x+1, find the opposite of each term.

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

Combine 2x and -x to get x.

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

Subtract 1 from -1 to get -2.

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

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

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

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

$$x-x^{2}-2=-2x+1-3x^{2}$$

Add 2x to both sides.

$$x-x^{2}-2+2x=1-3x^{2}$$

Combine x and 2x to get 3x.

$$3x-x^{2}-2=1-3x^{2}$$

Subtract 1 from both sides.

$$3x-x^{2}-2-1=-3x^{2}$$

Subtract 1 from -2 to get -3.

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

Add 3x^{2} to both sides.

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

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

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

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

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 2\left(-3\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-3±\sqrt{9-8\left(-3\right)}}{2\times 2}$$

Multiply -8 times -3.

$$x=\frac{-3±\sqrt{9+24}}{2\times 2}$$

Add 9 to 24.

$$x=\frac{-3±\sqrt{33}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-3±\sqrt{33}}{4}$$

Now solve the equation x=\frac{-3±\sqrt{33}}{4} when ± is plus. Add -3 to \sqrt{33}.

$$x=\frac{\sqrt{33}-3}{4}$$

Now solve the equation x=\frac{-3±\sqrt{33}}{4} when ± is minus. Subtract \sqrt{33} from -3.

$$x=\frac{-\sqrt{33}-3}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{33}-3}{4} x=\frac{-\sqrt{33}-3}{4}$$

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 \left(x-1\right)\left(x+1\right), the least common multiple of 1+x,1-x^{2},1-x.

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

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

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

To find the opposite of x+1, find the opposite of each term.

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

Combine 2x and -x to get x.

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

Subtract 1 from -1 to get -2.

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

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

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

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

$$x-x^{2}-2=-2x+1-3x^{2}$$

Add 2x to both sides.

$$x-x^{2}-2+2x=1-3x^{2}$$

Combine x and 2x to get 3x.

$$3x-x^{2}-2=1-3x^{2}$$

Add 3x^{2} to both sides.

$$3x-x^{2}-2+3x^{2}=1$$

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

$$3x+2x^{2}-2=1$$

Add 2 to both sides.

$$3x+2x^{2}=1+2$$

Add 1 and 2 to get 3.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by 2.

$$\frac{2x^{2}+3x}{2}=\frac{3}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{3}{2}x=\frac{3}{2}$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{3}{2}+\frac{9}{16}$$

Add \frac{3}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{33}{16}$$

Factor x^{2}+\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{33}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{33}{16}}$$

Simplify.

$$x+\frac{3}{4}=\frac{\sqrt{33}}{4} x+\frac{3}{4}=-\frac{\sqrt{33}}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$x=\frac{\sqrt{33}-3}{4} x=\frac{-\sqrt{33}-3}{4}$$