Solve x^2-36x+12=1

$$x ^ { 2 } - 36 x + 12 = 1$$

Solve for x

$$x=\sqrt{313}+18\approx 35.691806013$$

$$x=18-\sqrt{313}\approx 0.308193987$$

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.

$$x^{2}-36x+12=1$$

Subtract 1 from both sides of the equation.

$$x^{2}-36x+12-1=1-1$$

Subtracting 1 from itself leaves 0.

$$x^{2}-36x+12-1=0$$

Subtract 1 from 12.

$$x^{2}-36x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -36 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 11}}{2}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 11}}{2}$$

Multiply -4 times 11.

$$x=\frac{-\left(-36\right)±\sqrt{1296-44}}{2}$$

Add 1296 to -44.

$$x=\frac{-\left(-36\right)±\sqrt{1252}}{2}$$

Take the square root of 1252.

$$x=\frac{-\left(-36\right)±2\sqrt{313}}{2}$$

The opposite of -36 is 36.

$$x=\frac{36±2\sqrt{313}}{2}$$

Now solve the equation x=\frac{36±2\sqrt{313}}{2} when ± is plus. Add 36 to 2\sqrt{313}.

$$x=\frac{2\sqrt{313}+36}{2}$$

Divide 36+2\sqrt{313} by 2.

$$x=\sqrt{313}+18$$

Now solve the equation x=\frac{36±2\sqrt{313}}{2} when ± is minus. Subtract 2\sqrt{313} from 36.

$$x=\frac{36-2\sqrt{313}}{2}$$

Divide 36-2\sqrt{313} by 2.

$$x=18-\sqrt{313}$$

The equation is now solved.

$$x=\sqrt{313}+18 x=18-\sqrt{313}$$

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.

$$x^{2}-36x+12=1$$

Subtract 12 from both sides of the equation.

$$x^{2}-36x+12-12=1-12$$

Subtracting 12 from itself leaves 0.

$$x^{2}-36x=1-12$$

Subtract 12 from 1.

$$x^{2}-36x=-11$$

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.

$$x^{2}-36x+\left(-18\right)^{2}=-11+\left(-18\right)^{2}$$

Square -18.

$$x^{2}-36x+324=-11+324$$

Add -11 to 324.

$$x^{2}-36x+324=313$$

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}.

$$\left(x-18\right)^{2}=313$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-18\right)^{2}}=\sqrt{313}$$

Simplify.

$$x-18=\sqrt{313} x-18=-\sqrt{313}$$

Add 18 to both sides of the equation.

$$x=\sqrt{313}+18 x=18-\sqrt{313}$$