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

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

Solve for x

$$x=-2$$

Steps Using Factoring

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 x-1,1-x^{2}.

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

Use the distributive property to multiply x+1 by 2x-1 and combine like terms.

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

Subtract 2 from -1 to get -3.

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

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Subtract x^{2} from both sides.

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

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

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

Add 1 to both sides.

$$x^{2}+x-3+1=0$$

Add -3 and 1 to get -2.

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

To solve the equation, factor x^{2}+x-2 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=1 ab=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=2$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-1\right)\left(x+2\right)$$

To find equation solutions, solve x-1=0 and x+2=0.

$$x=1 x=-2$$

Variable x cannot be equal to 1.

$$x=-2$$

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

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

Use the distributive property to multiply x+1 by 2x-1 and combine like terms.

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

Subtract 2 from -1 to get -3.

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

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Subtract x^{2} from both sides.

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

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

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

Add 1 to both sides.

$$x^{2}+x-3+1=0$$

Add -3 and 1 to get -2.

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

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

$$a+b=1 ab=1\left(-2\right)=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=2$$

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

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

Factor out x in the first and 2 in the second group.

$$x\left(x-1\right)+2\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+2\right)$$

To find equation solutions, solve x-1=0 and x+2=0.

$$x=1 x=-2$$

Variable x cannot be equal to 1.

$$x=-2$$

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 x-1,1-x^{2}.

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

Use the distributive property to multiply x+1 by 2x-1 and combine like terms.

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

Subtract 2 from -1 to get -3.

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

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Subtract x^{2} from both sides.

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

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

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

Add 1 to both sides.

$$x^{2}+x-3+1=0$$

Add -3 and 1 to get -2.

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

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

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

Square 1.

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

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8}}{2}$$

Add 1 to 8.

$$x=\frac{-1±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-1±3}{2}$$

Now solve the equation x=\frac{-1±3}{2} when ± is plus. Add -1 to 3.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-1±3}{2} when ± is minus. Subtract 3 from -1.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=1 x=-2$$

Variable x cannot be equal to 1.

$$x=-2$$

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 x-1,1-x^{2}.

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

Use the distributive property to multiply x+1 by 2x-1 and combine like terms.

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

Subtract 2 from -1 to get -3.

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

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Subtract x^{2} from both sides.

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

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

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

Add 3 to both sides.

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

Add -1 and 3 to get 2.

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

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.

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

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

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

Add 2 to \frac{1}{4}.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

$$x=1 x=-2$$

Variable x cannot be equal to 1.

$$x=-2$$