Solve 6=-1/6x^2+2x+4

$$6 = - \frac { 1 } { 6 } x ^ { 2 } + 2 x + 4$$

Solve for x

$$x=2\sqrt{6}+6\approx 10.898979486$$

$$x=6-2\sqrt{6}\approx 1.101020514$$

Steps Using the Quadratic Formula

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

$$-\frac{1}{6}x^{2}+2x+4=6$$

Subtract 6 from both sides.

$$-\frac{1}{6}x^{2}+2x+4-6=0$$

Subtract 6 from 4 to get -2.

$$-\frac{1}{6}x^{2}+2x-2=0$$

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

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

Square 2.

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

Multiply -4 times -\frac{1}{6}.

$$x=\frac{-2±\sqrt{4+\frac{2}{3}\left(-2\right)}}{2\left(-\frac{1}{6}\right)}$$

Multiply \frac{2}{3} times -2.

$$x=\frac{-2±\sqrt{4-\frac{4}{3}}}{2\left(-\frac{1}{6}\right)}$$

Add 4 to -\frac{4}{3}.

$$x=\frac{-2±\sqrt{\frac{8}{3}}}{2\left(-\frac{1}{6}\right)}$$

Take the square root of \frac{8}{3}.

$$x=\frac{-2±\frac{2\sqrt{6}}{3}}{2\left(-\frac{1}{6}\right)}$$

Multiply 2 times -\frac{1}{6}.

$$x=\frac{-2±\frac{2\sqrt{6}}{3}}{-\frac{1}{3}}$$

Now solve the equation x=\frac{-2±\frac{2\sqrt{6}}{3}}{-\frac{1}{3}} when ± is plus. Add -2 to \frac{2\sqrt{6}}{3}.

$$x=\frac{\frac{2\sqrt{6}}{3}-2}{-\frac{1}{3}}$$

Divide -2+\frac{2\sqrt{6}}{3} by -\frac{1}{3} by multiplying -2+\frac{2\sqrt{6}}{3} by the reciprocal of -\frac{1}{3}.

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

Now solve the equation x=\frac{-2±\frac{2\sqrt{6}}{3}}{-\frac{1}{3}} when ± is minus. Subtract \frac{2\sqrt{6}}{3} from -2.

$$x=\frac{-\frac{2\sqrt{6}}{3}-2}{-\frac{1}{3}}$$

Divide -2-\frac{2\sqrt{6}}{3} by -\frac{1}{3} by multiplying -2-\frac{2\sqrt{6}}{3} by the reciprocal of -\frac{1}{3}.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$-\frac{1}{6}x^{2}+2x+4=6$$

Subtract 4 from both sides.

$$-\frac{1}{6}x^{2}+2x=6-4$$

Subtract 4 from 6 to get 2.

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

Multiply both sides by -6.

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

Dividing by -\frac{1}{6} undoes the multiplication by -\frac{1}{6}.

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

Divide 2 by -\frac{1}{6} by multiplying 2 by the reciprocal of -\frac{1}{6}.

$$x^{2}-12x=\frac{2}{-\frac{1}{6}}$$

Divide 2 by -\frac{1}{6} by multiplying 2 by the reciprocal of -\frac{1}{6}.

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

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.

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

Square -6.

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

Add -12 to 36.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 6 to both sides of the equation.

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