Solve x^2-36x+72=0

$${ x }^{ 2 } -36x+72=0$$

Solve for x

$$x=6\sqrt{7}+18\approx 33.874507866$$

$$x=18-6\sqrt{7}\approx 2.125492134$$

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.

$$x^{2}-36x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -36 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 72}}{2}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 72}}{2}$$

Multiply -4 times 72.

$$x=\frac{-\left(-36\right)±\sqrt{1296-288}}{2}$$

Add 1296 to -288.

$$x=\frac{-\left(-36\right)±\sqrt{1008}}{2}$$

Take the square root of 1008.

$$x=\frac{-\left(-36\right)±12\sqrt{7}}{2}$$

The opposite of -36 is 36.

$$x=\frac{36±12\sqrt{7}}{2}$$

Now solve the equation x=\frac{36±12\sqrt{7}}{2} when ± is plus. Add 36 to 12\sqrt{7}.

$$x=\frac{12\sqrt{7}+36}{2}$$

Divide 36+12\sqrt{7} by 2.

$$x=6\sqrt{7}+18$$

Now solve the equation x=\frac{36±12\sqrt{7}}{2} when ± is minus. Subtract 12\sqrt{7} from 36.

$$x=\frac{36-12\sqrt{7}}{2}$$

Divide 36-12\sqrt{7} by 2.

$$x=18-6\sqrt{7}$$

The equation is now solved.

$$x=6\sqrt{7}+18 x=18-6\sqrt{7}$$

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.

$$x^{2}-36x+72=0$$

Subtract 72 from both sides of the equation.

$$x^{2}-36x+72-72=-72$$

Subtracting 72 from itself leaves 0.

$$x^{2}-36x=-72$$

Divide -36, the coefficient of the x term, by 2 to get -18. Then add the square of -18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-36x+\left(-18\right)^{2}=-72+\left(-18\right)^{2}$$

Square -18.

$$x^{2}-36x+324=-72+324$$

Add -72 to 324.

$$x^{2}-36x+324=252$$

Factor x^{2}-36x+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-18\right)^{2}=252$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-18\right)^{2}}=\sqrt{252}$$

Simplify.

$$x-18=6\sqrt{7} x-18=-6\sqrt{7}$$

Add 18 to both sides of the equation.

$$x=6\sqrt{7}+18 x=18-6\sqrt{7}$$