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

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

Solve for x

$$x=\frac{1}{8}=0.125$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

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

$$x=8x^{2}-8x+1$$

Subtract 8x^{2} from both sides.

$$x-8x^{2}=-8x+1$$

Add 8x to both sides.

$$x-8x^{2}+8x=1$$

Combine x and 8x to get 9x.

$$9x-8x^{2}=1$$

Subtract 1 from both sides.

$$9x-8x^{2}-1=0$$

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.

$$-8x^{2}+9x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 9 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-8\right)\left(-1\right)}}{2\left(-8\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-8\right)\left(-1\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-9±\sqrt{81+32\left(-1\right)}}{2\left(-8\right)}$$

Multiply 32 times -1.

$$x=\frac{-9±\sqrt{81-32}}{2\left(-8\right)}$$

Add 81 to -32.

$$x=\frac{-9±\sqrt{49}}{2\left(-8\right)}$$

Take the square root of 49.

$$x=\frac{-9±7}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-9±7}{-16}$$

Now solve the equation x=\frac{-9±7}{-16} when ± is plus. Add -9 to 7.

$$x=-\frac{2}{-16}$$

Reduce the fraction \frac{-2}{-16} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-9±7}{-16} when ± is minus. Subtract 7 from -9.

$$x=-\frac{16}{-16}$$

Divide -16 by -16.

$$x=1$$

The equation is now solved.

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

Variable x cannot be equal to 1.

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

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

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

$$x=8x^{2}-8x+1$$

Subtract 8x^{2} from both sides.

$$x-8x^{2}=-8x+1$$

Add 8x to both sides.

$$x-8x^{2}+8x=1$$

Combine x and 8x to get 9x.

$$9x-8x^{2}=1$$

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.

$$-8x^{2}+9x=1$$

Divide both sides by -8.

$$\frac{-8x^{2}+9x}{-8}=\frac{1}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{9}{-8}x=\frac{1}{-8}$$

Divide 9 by -8.

$$x^{2}-\frac{9}{8}x=\frac{1}{-8}$$

Divide 1 by -8.

$$x^{2}-\frac{9}{8}x=-\frac{1}{8}$$

Divide -\frac{9}{8}, the coefficient of the x term, by 2 to get -\frac{9}{16}. Then add the square of -\frac{9}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{8}x+\left(-\frac{9}{16}\right)^{2}=-\frac{1}{8}+\left(-\frac{9}{16}\right)^{2}$$

Square -\frac{9}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{8}x+\frac{81}{256}=-\frac{1}{8}+\frac{81}{256}$$

Add -\frac{1}{8} to \frac{81}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{8}x+\frac{81}{256}=\frac{49}{256}$$

Factor x^{2}-\frac{9}{8}x+\frac{81}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{16}\right)^{2}=\frac{49}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{16}\right)^{2}}=\sqrt{\frac{49}{256}}$$

Simplify.

$$x-\frac{9}{16}=\frac{7}{16} x-\frac{9}{16}=-\frac{7}{16}$$

Add \frac{9}{16} to both sides of the equation.

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

Variable x cannot be equal to 1.

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