Solve x^2+3x/x^2-1-5x/x+1=x/x-1

$$\frac { x ^ { 2 } + 3 x } { x ^ { 2 } - 1 } - \frac { 5 x } { x + 1 } = \frac { x } { x - 1 }$$

Solve for x

$$x = \frac{7}{5} = 1\frac{2}{5} = 1.4$$

$$x=0$$

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

$$x^{2}+3x-\left(x-1\right)\times 5x=\left(x+1\right)x$$

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

$$x^{2}+3x-\left(5x-5\right)x=\left(x+1\right)x$$

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

$$x^{2}+3x-\left(5x^{2}-5x\right)=\left(x+1\right)x$$

To find the opposite of 5x^{2}-5x, find the opposite of each term.

$$x^{2}+3x-5x^{2}+5x=\left(x+1\right)x$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+3x+5x=\left(x+1\right)x$$

Combine 3x and 5x to get 8x.

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

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

$$-4x^{2}+8x=x^{2}+x$$

Subtract x^{2} from both sides.

$$-4x^{2}+8x-x^{2}=x$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$-5x^{2}+8x=x$$

Subtract x from both sides.

$$-5x^{2}+8x-x=0$$

Combine 8x and -x to get 7x.

$$-5x^{2}+7x=0$$

Factor out x.

$$x\left(-5x+7\right)=0$$

To find equation solutions, solve x=0 and -5x+7=0.

$$x=0 x=\frac{7}{5}$$

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

$$x^{2}+3x-\left(x-1\right)\times 5x=\left(x+1\right)x$$

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

$$x^{2}+3x-\left(5x-5\right)x=\left(x+1\right)x$$

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

$$x^{2}+3x-\left(5x^{2}-5x\right)=\left(x+1\right)x$$

To find the opposite of 5x^{2}-5x, find the opposite of each term.

$$x^{2}+3x-5x^{2}+5x=\left(x+1\right)x$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+3x+5x=\left(x+1\right)x$$

Combine 3x and 5x to get 8x.

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

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

$$-4x^{2}+8x=x^{2}+x$$

Subtract x^{2} from both sides.

$$-4x^{2}+8x-x^{2}=x$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$-5x^{2}+8x=x$$

Subtract x from both sides.

$$-5x^{2}+8x-x=0$$

Combine 8x and -x to get 7x.

$$-5x^{2}+7x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 7 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}}}{2\left(-5\right)}$$

Take the square root of 7^{2}.

$$x=\frac{-7±7}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-7±7}{-10}$$

Now solve the equation x=\frac{-7±7}{-10} when ± is plus. Add -7 to 7.

$$x=\frac{0}{-10}$$

Divide 0 by -10.

$$x=0$$

Now solve the equation x=\frac{-7±7}{-10} when ± is minus. Subtract 7 from -7.

$$x=-\frac{14}{-10}$$

Reduce the fraction \frac{-14}{-10} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{5}$$

The equation is now solved.

$$x=0 x=\frac{7}{5}$$

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

$$x^{2}+3x-\left(x-1\right)\times 5x=\left(x+1\right)x$$

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

$$x^{2}+3x-\left(5x-5\right)x=\left(x+1\right)x$$

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

$$x^{2}+3x-\left(5x^{2}-5x\right)=\left(x+1\right)x$$

To find the opposite of 5x^{2}-5x, find the opposite of each term.

$$x^{2}+3x-5x^{2}+5x=\left(x+1\right)x$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+3x+5x=\left(x+1\right)x$$

Combine 3x and 5x to get 8x.

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

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

$$-4x^{2}+8x=x^{2}+x$$

Subtract x^{2} from both sides.

$$-4x^{2}+8x-x^{2}=x$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$-5x^{2}+8x=x$$

Subtract x from both sides.

$$-5x^{2}+8x-x=0$$

Combine 8x and -x to get 7x.

$$-5x^{2}+7x=0$$

Divide both sides by -5.

$$\frac{-5x^{2}+7x}{-5}=\frac{0}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{7}{-5}x=\frac{0}{-5}$$

Divide 7 by -5.

$$x^{2}-\frac{7}{5}x=\frac{0}{-5}$$

Divide 0 by -5.

$$x^{2}-\frac{7}{5}x=0$$

Divide -\frac{7}{5}, the coefficient of the x term, by 2 to get -\frac{7}{10}. Then add the square of -\frac{7}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{5}x+\left(-\frac{7}{10}\right)^{2}=\left(-\frac{7}{10}\right)^{2}$$

Square -\frac{7}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{49}{100}$$

Factor x^{2}-\frac{7}{5}x+\frac{49}{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}.

$$\left(x-\frac{7}{10}\right)^{2}=\frac{49}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Simplify.

$$x-\frac{7}{10}=\frac{7}{10} x-\frac{7}{10}=-\frac{7}{10}$$

Add \frac{7}{10} to both sides of the equation.

$$x=\frac{7}{5} x=0$$