Solve -x/1-x=8x/x+1

$$\frac { - x } { 1 - x } = \frac { 8 x } { x + 1 }$$

Solve for x

$$x = \frac{9}{7} = 1\frac{2}{7} \approx 1.285714286$$

$$x=0$$

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 1-x,x+1.

$$-\left(1+x\right)\left(-x\right)=\left(x-1\right)\times 8x$$

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

$$\left(-1-x\right)\left(-x\right)=\left(x-1\right)\times 8x$$

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

$$-\left(-x\right)-x\left(-x\right)=\left(x-1\right)\times 8x$$

Multiply -1 and -1 to get 1.

$$x-x\left(-x\right)=\left(x-1\right)\times 8x$$

Multiply -1 and -1 to get 1.

$$x+xx=\left(x-1\right)\times 8x$$

Multiply x and x to get x^{2}.

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

Use the distributive property to multiply x-1 by 8.

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

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

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

Subtract 8x^{2} from both sides.

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

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$x-7x^{2}=-8x$$

Add 8x to both sides.

$$x-7x^{2}+8x=0$$

Combine x and 8x to get 9x.

$$9x-7x^{2}=0$$

Factor out x.

$$x\left(9-7x\right)=0$$

To find equation solutions, solve x=0 and 9-7x=0.

$$x=0 x=\frac{9}{7}$$

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,x+1.

$$-\left(1+x\right)\left(-x\right)=\left(x-1\right)\times 8x$$

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

$$\left(-1-x\right)\left(-x\right)=\left(x-1\right)\times 8x$$

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

$$-\left(-x\right)-x\left(-x\right)=\left(x-1\right)\times 8x$$

Multiply -1 and -1 to get 1.

$$x-x\left(-x\right)=\left(x-1\right)\times 8x$$

Multiply -1 and -1 to get 1.

$$x+xx=\left(x-1\right)\times 8x$$

Multiply x and x to get x^{2}.

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

Use the distributive property to multiply x-1 by 8.

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

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

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

Subtract 8x^{2} from both sides.

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

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$x-7x^{2}=-8x$$

Add 8x to both sides.

$$x-7x^{2}+8x=0$$

Combine x and 8x to get 9x.

$$9x-7x^{2}=0$$

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.

$$-7x^{2}+9x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}}}{2\left(-7\right)}$$

Take the square root of 9^{2}.

$$x=\frac{-9±9}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-9±9}{-14}$$

Now solve the equation x=\frac{-9±9}{-14} when ± is plus. Add -9 to 9.

$$x=\frac{0}{-14}$$

Divide 0 by -14.

$$x=0$$

Now solve the equation x=\frac{-9±9}{-14} when ± is minus. Subtract 9 from -9.

$$x=-\frac{18}{-14}$$

Reduce the fraction \frac{-18}{-14} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{7}$$

The equation is now solved.

$$x=0 x=\frac{9}{7}$$

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,x+1.

$$-\left(1+x\right)\left(-x\right)=\left(x-1\right)\times 8x$$

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

$$\left(-1-x\right)\left(-x\right)=\left(x-1\right)\times 8x$$

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

$$-\left(-x\right)-x\left(-x\right)=\left(x-1\right)\times 8x$$

Multiply -1 and -1 to get 1.

$$x-x\left(-x\right)=\left(x-1\right)\times 8x$$

Multiply -1 and -1 to get 1.

$$x+xx=\left(x-1\right)\times 8x$$

Multiply x and x to get x^{2}.

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

Use the distributive property to multiply x-1 by 8.

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

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

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

Subtract 8x^{2} from both sides.

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

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$x-7x^{2}=-8x$$

Add 8x to both sides.

$$x-7x^{2}+8x=0$$

Combine x and 8x to get 9x.

$$9x-7x^{2}=0$$

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.

$$-7x^{2}+9x=0$$

Divide both sides by -7.

$$\frac{-7x^{2}+9x}{-7}=\frac{0}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{9}{-7}x=\frac{0}{-7}$$

Divide 9 by -7.

$$x^{2}-\frac{9}{7}x=\frac{0}{-7}$$

Divide 0 by -7.

$$x^{2}-\frac{9}{7}x=0$$

Divide -\frac{9}{7}, the coefficient of the x term, by 2 to get -\frac{9}{14}. Then add the square of -\frac{9}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{7}x+\left(-\frac{9}{14}\right)^{2}=\left(-\frac{9}{14}\right)^{2}$$

Square -\frac{9}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{7}x+\frac{81}{196}=\frac{81}{196}$$

Factor x^{2}-\frac{9}{7}x+\frac{81}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{14}\right)^{2}=\frac{81}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{14}\right)^{2}}=\sqrt{\frac{81}{196}}$$

Simplify.

$$x-\frac{9}{14}=\frac{9}{14} x-\frac{9}{14}=-\frac{9}{14}$$

Add \frac{9}{14} to both sides of the equation.

$$x=\frac{9}{7} x=0$$