Solve 0=x^2-3x+3

$$0 = x ^ { 2 } - 3 x + 3$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{3}i}{2}\approx 1.5+0.866025404i$$

$$x=\frac{-\sqrt{3}i+3}{2}\approx 1.5-0.866025404i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-3x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 3}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-\left(-3\right)±\sqrt{9-12}}{2}$$

Add 9 to -12.

$$x=\frac{-\left(-3\right)±\sqrt{-3}}{2}$$

Take the square root of -3.

$$x=\frac{-\left(-3\right)±\sqrt{3}i}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{3}i}{2}$$

Now solve the equation x=\frac{3±\sqrt{3}i}{2} when ± is plus. Add 3 to i\sqrt{3}.

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

Now solve the equation x=\frac{3±\sqrt{3}i}{2} when ± is minus. Subtract i\sqrt{3} from 3.

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

The equation is now solved.

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

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-3x+3=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-3x=-3$$

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

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-3+\frac{9}{4}$$

Add -3 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=-\frac{3}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

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

Add \frac{3}{2} to both sides of the equation.

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