Solve 108x^2+12=72x

$$108 x ^ { 2 } + 12 = 72 x$$

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using the Quadratic Formula

Subtract 72x from both sides.

$$108x^{2}+12-72x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$108x^{2}-72x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 108 for a, -72 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 108\times 12}}{2\times 108}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 108\times 12}}{2\times 108}$$

Multiply -4 times 108.

$$x=\frac{-\left(-72\right)±\sqrt{5184-432\times 12}}{2\times 108}$$

Multiply -432 times 12.

$$x=\frac{-\left(-72\right)±\sqrt{5184-5184}}{2\times 108}$$

Add 5184 to -5184.

$$x=\frac{-\left(-72\right)±\sqrt{0}}{2\times 108}$$

Take the square root of 0.

$$x=-\frac{-72}{2\times 108}$$

The opposite of -72 is 72.

$$x=\frac{72}{2\times 108}$$

Multiply 2 times 108.

$$x=\frac{72}{216}$$

Reduce the fraction \frac{72}{216} to lowest terms by extracting and canceling out 72.

$$x=\frac{1}{3}$$

Steps for Completing the Square

Subtract 72x from both sides.

$$108x^{2}+12-72x=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$108x^{2}-72x=-12$$

Divide both sides by 108.

$$\frac{108x^{2}-72x}{108}=-\frac{12}{108}$$

Dividing by 108 undoes the multiplication by 108.

$$x^{2}+\left(-\frac{72}{108}\right)x=-\frac{12}{108}$$

Reduce the fraction \frac{-72}{108} to lowest terms by extracting and canceling out 36.

$$x^{2}-\frac{2}{3}x=-\frac{12}{108}$$

Reduce the fraction \frac{-12}{108} to lowest terms by extracting and canceling out 12.

$$x^{2}-\frac{2}{3}x=-\frac{1}{9}$$

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=-\frac{1}{9}+\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{-1+1}{9}$$

Add -\frac{1}{9} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=0$$

Factor x^{2}-\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{3}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{1}{3}=0 x-\frac{1}{3}=0$$

Add \frac{1}{3} to both sides of the equation.

$$x=\frac{1}{3} x=\frac{1}{3}$$

The equation is now solved. Solutions are the same.

$$x=\frac{1}{3}$$