Solve 36x=13+36x^2

$$36x = 13+36 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{1}{2}-\frac{1}{3}i\approx 0.5-0.333333333i$$

$$x=\frac{1}{2}+\frac{1}{3}i\approx 0.5+0.333333333i$$

Steps Using the Quadratic Formula

Subtract 13 from both sides.

$$36x-13=36x^{2}$$

Subtract 36x^{2} from both sides.

$$36x-13-36x^{2}=0$$

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.

$$-36x^{2}+36x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -36 for a, 36 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\left(-36\right)\left(-13\right)}}{2\left(-36\right)}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\left(-36\right)\left(-13\right)}}{2\left(-36\right)}$$

Multiply -4 times -36.

$$x=\frac{-36±\sqrt{1296+144\left(-13\right)}}{2\left(-36\right)}$$

Multiply 144 times -13.

$$x=\frac{-36±\sqrt{1296-1872}}{2\left(-36\right)}$$

Add 1296 to -1872.

$$x=\frac{-36±\sqrt{-576}}{2\left(-36\right)}$$

Take the square root of -576.

$$x=\frac{-36±24i}{2\left(-36\right)}$$

Multiply 2 times -36.

$$x=\frac{-36±24i}{-72}$$

Now solve the equation x=\frac{-36±24i}{-72} when ± is plus. Add -36 to 24i.

$$x=\frac{-36+24i}{-72}$$

Divide -36+24i by -72.

$$x=\frac{1}{2}-\frac{1}{3}i$$

Now solve the equation x=\frac{-36±24i}{-72} when ± is minus. Subtract 24i from -36.

$$x=\frac{-36-24i}{-72}$$

Divide -36-24i by -72.

$$x=\frac{1}{2}+\frac{1}{3}i$$

The equation is now solved.

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

Steps for Completing the Square

Subtract 36x^{2} from both sides.

$$36x-36x^{2}=13$$

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.

$$-36x^{2}+36x=13$$

Divide both sides by -36.

$$\frac{-36x^{2}+36x}{-36}=\frac{13}{-36}$$

Dividing by -36 undoes the multiplication by -36.

$$x^{2}+\frac{36}{-36}x=\frac{13}{-36}$$

Divide 36 by -36.

$$x^{2}-x=\frac{13}{-36}$$

Divide 13 by -36.

$$x^{2}-x=-\frac{13}{36}$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{13}{36}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=-\frac{13}{36}+\frac{1}{4}$$

Add -\frac{13}{36} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=-\frac{1}{9}$$

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

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{1}{2}=\frac{1}{3}i x-\frac{1}{2}=-\frac{1}{3}i$$

Add \frac{1}{2} to both sides of the equation.

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