Solve x^2-6x=6x-24

$${ x }^{ 2 } -6x = 6x-24$$

Solve for x

$$x=2\sqrt{3}+6\approx 9.464101615$$

$$x=6-2\sqrt{3}\approx 2.535898385$$

Steps Using the Quadratic Formula

Subtract 6x from both sides.

$$x^{2}-6x-6x=-24$$

Combine -6x and -6x to get -12x.

$$x^{2}-12x=-24$$

Add 24 to both sides.

$$x^{2}-12x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 24}}{2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 24}}{2}$$

Multiply -4 times 24.

$$x=\frac{-\left(-12\right)±\sqrt{144-96}}{2}$$

Add 144 to -96.

$$x=\frac{-\left(-12\right)±\sqrt{48}}{2}$$

Take the square root of 48.

$$x=\frac{-\left(-12\right)±4\sqrt{3}}{2}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{3}}{2}$$

Now solve the equation x=\frac{12±4\sqrt{3}}{2} when ± is plus. Add 12 to 4\sqrt{3}.

$$x=\frac{4\sqrt{3}+12}{2}$$

Divide 12+4\sqrt{3} by 2.

$$x=2\sqrt{3}+6$$

Now solve the equation x=\frac{12±4\sqrt{3}}{2} when ± is minus. Subtract 4\sqrt{3} from 12.

$$x=\frac{12-4\sqrt{3}}{2}$$

Divide 12-4\sqrt{3} by 2.

$$x=6-2\sqrt{3}$$

The equation is now solved.

$$x=2\sqrt{3}+6 x=6-2\sqrt{3}$$

Steps for Completing the Square

Subtract 6x from both sides.

$$x^{2}-6x-6x=-24$$

Combine -6x and -6x to get -12x.

$$x^{2}-12x=-24$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-24+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-24+36$$

Add -24 to 36.

$$x^{2}-12x+36=12$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{12}$$

Simplify.

$$x-6=2\sqrt{3} x-6=-2\sqrt{3}$$

Add 6 to both sides of the equation.

$$x=2\sqrt{3}+6 x=6-2\sqrt{3}$$