Solve 12/6-x=x+1

$$\frac { 12 } { 6 - x } = x + 1$$

Solve for x

$$x=2$$

$$x=3$$

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.

$$12=\left(-x+6\right)x-x+6$$

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

$$12=-x^{2}+6x-x+6$$

Combine 6x and -x to get 5x.

$$12=-x^{2}+5x+6$$

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

$$-x^{2}+5x+6=12$$

Subtract 12 from both sides.

$$-x^{2}+5x+6-12=0$$

Subtract 12 from 6 to get -6.

$$-x^{2}+5x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\left(-6\right)}}{2\left(-1\right)}$$

Multiply 4 times -6.

$$x=\frac{-5±\sqrt{25-24}}{2\left(-1\right)}$$

Add 25 to -24.

$$x=\frac{-5±\sqrt{1}}{2\left(-1\right)}$$

Take the square root of 1.

$$x=\frac{-5±1}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±1}{-2}$$

Now solve the equation x=\frac{-5±1}{-2} when ± is plus. Add -5 to 1.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-5±1}{-2} when ± is minus. Subtract 1 from -5.

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

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=2 x=3$$

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.

$$12=\left(-x+6\right)x-x+6$$

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

$$12=-x^{2}+6x-x+6$$

Combine 6x and -x to get 5x.

$$12=-x^{2}+5x+6$$

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

$$-x^{2}+5x+6=12$$

Subtract 6 from both sides.

$$-x^{2}+5x=12-6$$

Subtract 6 from 12 to get 6.

$$-x^{2}+5x=6$$

Divide both sides by -1.

$$\frac{-x^{2}+5x}{-1}=\frac{6}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{5}{-1}x=\frac{6}{-1}$$

Divide 5 by -1.

$$x^{2}-5x=\frac{6}{-1}$$

Divide 6 by -1.

$$x^{2}-5x=-6$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-6+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-6+\frac{25}{4}$$

Add -6 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{1}{4}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{5}{2}=\frac{1}{2} x-\frac{5}{2}=-\frac{1}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=3 x=2$$