Solve -x-8=33/x-6

$$- x - 8 = \frac { 33 } { x - 6 }$$

Solve for x

$$x=-5$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by x-6.

$$-\left(x-6\right)x+\left(x-6\right)\left(-8\right)=33$$

Use the distributive property to multiply x-6 by x.

$$-\left(x^{2}-6x\right)+\left(x-6\right)\left(-8\right)=33$$

To find the opposite of x^{2}-6x, find the opposite of each term.

$$-x^{2}+6x+\left(x-6\right)\left(-8\right)=33$$

Use the distributive property to multiply x-6 by -8.

$$-x^{2}+6x-8x+48=33$$

Combine 6x and -8x to get -2x.

$$-x^{2}-2x+48=33$$

Subtract 33 from both sides.

$$-x^{2}-2x+48-33=0$$

Subtract 33 from 48 to get 15.

$$-x^{2}-2x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 15}}{2\left(-1\right)}$$

Multiply 4 times 15.

$$x=\frac{-\left(-2\right)±\sqrt{4+60}}{2\left(-1\right)}$$

Add 4 to 60.

$$x=\frac{-\left(-2\right)±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

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

The opposite of -2 is 2.

$$x=\frac{2±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±8}{-2}$$

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

$$x=\frac{10}{-2}$$

Divide 10 by -2.

$$x=-5$$

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

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=-5 x=3$$

Steps for Completing the Square

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by x-6.

$$-\left(x-6\right)x+\left(x-6\right)\left(-8\right)=33$$

Use the distributive property to multiply x-6 by x.

$$-\left(x^{2}-6x\right)+\left(x-6\right)\left(-8\right)=33$$

To find the opposite of x^{2}-6x, find the opposite of each term.

$$-x^{2}+6x+\left(x-6\right)\left(-8\right)=33$$

Use the distributive property to multiply x-6 by -8.

$$-x^{2}+6x-8x+48=33$$

Combine 6x and -8x to get -2x.

$$-x^{2}-2x+48=33$$

Subtract 48 from both sides.

$$-x^{2}-2x=33-48$$

Subtract 48 from 33 to get -15.

$$-x^{2}-2x=-15$$

Divide both sides by -1.

$$\frac{-x^{2}-2x}{-1}=-\frac{15}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{2}{-1}\right)x=-\frac{15}{-1}$$

Divide -2 by -1.

$$x^{2}+2x=-\frac{15}{-1}$$

Divide -15 by -1.

$$x^{2}+2x=15$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=15+1^{2}$$

Square 1.

$$x^{2}+2x+1=15+1$$

Add 15 to 1.

$$x^{2}+2x+1=16$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+1=4 x+1=-4$$

Subtract 1 from both sides of the equation.

$$x=3 x=-5$$