Solve 9/x-8=x

$$\frac { 9 } { x - 8 } = x$$

Solve for x

$$x=-1$$

$$x=9$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by x-8.

$$9=x\left(x-8\right)$$

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

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 9 from both sides.

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

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

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

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$x=\frac{-\left(-8\right)±\sqrt{64+36}}{2}$$

Add 64 to 36.

$$x=\frac{-\left(-8\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-8\right)±10}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±10}{2}$$

Now solve the equation x=\frac{8±10}{2} when ± is plus. Add 8 to 10.

$$x=\frac{18}{2}$$

Divide 18 by 2.

$$x=9$$

Now solve the equation x=\frac{8±10}{2} when ± is minus. Subtract 10 from 8.

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=9 x=-1$$

Steps for Completing the Square

Variable x cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by x-8.

$$9=x\left(x-8\right)$$

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

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

Swap sides so that all variable terms are on the left hand side.

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

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=9+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=9+16$$

Add 9 to 16.

$$x^{2}-8x+16=25$$

Factor x^{2}-8x+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}.

$$\left(x-4\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-4=5 x-4=-5$$

Add 4 to both sides of the equation.

$$x=9 x=-1$$