Solve 72x^2-24x+1=0

$$72 { x }^{ 2 } -24x+1=0$$

Solve for x

$$x=\frac{\sqrt{2}}{12}+\frac{1}{6}\approx 0.284517797$$

$$x=-\frac{\sqrt{2}}{12}+\frac{1}{6}\approx 0.048815536$$

Steps Using the Quadratic Formula

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.

$$72x^{2}-24x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 72 for a, -24 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 72}}{2\times 72}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 72}}{2\times 72}$$

Multiply -4 times 72.

$$x=\frac{-\left(-24\right)±\sqrt{576-288}}{2\times 72}$$

Add 576 to -288.

$$x=\frac{-\left(-24\right)±\sqrt{288}}{2\times 72}$$

Take the square root of 288.

$$x=\frac{-\left(-24\right)±12\sqrt{2}}{2\times 72}$$

The opposite of -24 is 24.

$$x=\frac{24±12\sqrt{2}}{2\times 72}$$

Multiply 2 times 72.

$$x=\frac{24±12\sqrt{2}}{144}$$

Now solve the equation x=\frac{24±12\sqrt{2}}{144} when ± is plus. Add 24 to 12\sqrt{2}.

$$x=\frac{12\sqrt{2}+24}{144}$$

Divide 24+12\sqrt{2} by 144.

$$x=\frac{\sqrt{2}}{12}+\frac{1}{6}$$

Now solve the equation x=\frac{24±12\sqrt{2}}{144} when ± is minus. Subtract 12\sqrt{2} from 24.

$$x=\frac{24-12\sqrt{2}}{144}$$

Divide 24-12\sqrt{2} by 144.

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

The equation is now solved.

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

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$72x^{2}-24x+1=0$$

Subtract 1 from both sides of the equation.

$$72x^{2}-24x+1-1=-1$$

Subtracting 1 from itself leaves 0.

$$72x^{2}-24x=-1$$

Divide both sides by 72.

$$\frac{72x^{2}-24x}{72}=-\frac{1}{72}$$

Dividing by 72 undoes the multiplication by 72.

$$x^{2}+\left(-\frac{24}{72}\right)x=-\frac{1}{72}$$

Reduce the fraction \frac{-24}{72} to lowest terms by extracting and canceling out 24.

$$x^{2}-\frac{1}{3}x=-\frac{1}{72}$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=-\frac{1}{72}+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=-\frac{1}{72}+\frac{1}{36}$$

Add -\frac{1}{72} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{1}{72}$$

Factor x^{2}-\frac{1}{3}x+\frac{1}{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}.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{1}{72}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{1}{72}}$$

Simplify.

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

Add \frac{1}{6} to both sides of the equation.

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