Solve 3=1/3(12x^2-48x+36

$$3 = 1 / 3 ( 12 x ^ { 2 } - 48 x + 36$$

Solve for x

$$x=\frac{\sqrt{7}}{2}+2\approx 3.322875656$$

$$x=-\frac{\sqrt{7}}{2}+2\approx 0.677124344$$

Steps Using the Quadratic Formula

Multiply both sides by 3, the reciprocal of \frac{1}{3}.

$$3\times 3=12x^{2}-48x+36$$

Multiply 3 and 3 to get 9.

$$9=12x^{2}-48x+36$$

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

$$12x^{2}-48x+36=9$$

Subtract 9 from both sides.

$$12x^{2}-48x+36-9=0$$

Subtract 9 from 36 to get 27.

$$12x^{2}-48x+27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -48 for b, and 27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 12\times 27}}{2\times 12}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 12\times 27}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-48\right)±\sqrt{2304-48\times 27}}{2\times 12}$$

Multiply -48 times 27.

$$x=\frac{-\left(-48\right)±\sqrt{2304-1296}}{2\times 12}$$

Add 2304 to -1296.

$$x=\frac{-\left(-48\right)±\sqrt{1008}}{2\times 12}$$

Take the square root of 1008.

$$x=\frac{-\left(-48\right)±12\sqrt{7}}{2\times 12}$$

The opposite of -48 is 48.

$$x=\frac{48±12\sqrt{7}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{48±12\sqrt{7}}{24}$$

Now solve the equation x=\frac{48±12\sqrt{7}}{24} when ± is plus. Add 48 to 12\sqrt{7}.

$$x=\frac{12\sqrt{7}+48}{24}$$

Divide 48+12\sqrt{7} by 24.

$$x=\frac{\sqrt{7}}{2}+2$$

Now solve the equation x=\frac{48±12\sqrt{7}}{24} when ± is minus. Subtract 12\sqrt{7} from 48.

$$x=\frac{48-12\sqrt{7}}{24}$$

Divide 48-12\sqrt{7} by 24.

$$x=-\frac{\sqrt{7}}{2}+2$$

The equation is now solved.

$$x=\frac{\sqrt{7}}{2}+2 x=-\frac{\sqrt{7}}{2}+2$$

Steps for Completing the Square

Multiply both sides by 3, the reciprocal of \frac{1}{3}.

$$3\times 3=12x^{2}-48x+36$$

Multiply 3 and 3 to get 9.

$$9=12x^{2}-48x+36$$

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

$$12x^{2}-48x+36=9$$

Subtract 36 from both sides.

$$12x^{2}-48x=9-36$$

Subtract 36 from 9 to get -27.

$$12x^{2}-48x=-27$$

Divide both sides by 12.

$$\frac{12x^{2}-48x}{12}=-\frac{27}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{48}{12}\right)x=-\frac{27}{12}$$

Divide -48 by 12.

$$x^{2}-4x=-\frac{27}{12}$$

Reduce the fraction \frac{-27}{12} to lowest terms by extracting and canceling out 3.

$$x^{2}-4x=-\frac{9}{4}$$

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

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

Square -2.

$$x^{2}-4x+4=-\frac{9}{4}+4$$

Add -\frac{9}{4} to 4.

$$x^{2}-4x+4=\frac{7}{4}$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{7}{4}}$$

Simplify.

$$x-2=\frac{\sqrt{7}}{2} x-2=-\frac{\sqrt{7}}{2}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{7}}{2}+2 x=-\frac{\sqrt{7}}{2}+2$$