Solve 18x=9x^2+45

$$18 x = 9 x ^ { 2 } + 45$$

Solve for x (complex solution)

$$x=1+2i$$

$$x=1-2i$$

Steps Using the Quadratic Formula

Subtract 9x^{2} from both sides.

$$18x-9x^{2}=45$$

Subtract 45 from both sides.

$$18x-9x^{2}-45=0$$

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.

$$-9x^{2}+18x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 18 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-9\right)\left(-45\right)}}{2\left(-9\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-9\right)\left(-45\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-18±\sqrt{324+36\left(-45\right)}}{2\left(-9\right)}$$

Multiply 36 times -45.

$$x=\frac{-18±\sqrt{324-1620}}{2\left(-9\right)}$$

Add 324 to -1620.

$$x=\frac{-18±\sqrt{-1296}}{2\left(-9\right)}$$

Take the square root of -1296.

$$x=\frac{-18±36i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-18±36i}{-18}$$

Now solve the equation x=\frac{-18±36i}{-18} when ± is plus. Add -18 to 36i.

$$x=\frac{-18+36i}{-18}$$

Divide -18+36i by -18.

$$x=1-2i$$

Now solve the equation x=\frac{-18±36i}{-18} when ± is minus. Subtract 36i from -18.

$$x=\frac{-18-36i}{-18}$$

Divide -18-36i by -18.

$$x=1+2i$$

The equation is now solved.

$$x=1-2i x=1+2i$$

Steps for Completing the Square

Subtract 9x^{2} from both sides.

$$18x-9x^{2}=45$$

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.

$$-9x^{2}+18x=45$$

Divide both sides by -9.

$$\frac{-9x^{2}+18x}{-9}=\frac{45}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{18}{-9}x=\frac{45}{-9}$$

Divide 18 by -9.

$$x^{2}-2x=\frac{45}{-9}$$

Divide 45 by -9.

$$x^{2}-2x=-5$$

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

$$x^{2}-2x+1=-5+1$$

Add -5 to 1.

$$x^{2}-2x+1=-4$$

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

$$\left(x-1\right)^{2}=-4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-4}$$

Simplify.

$$x-1=2i x-1=-2i$$

Add 1 to both sides of the equation.

$$x=1+2i x=1-2i$$