Solve 0=x^2-6x+21

$$0 = x ^ { 2 } - 6 x + 21$$

Solve for x (complex solution)

$$x=3+2\sqrt{3}i\approx 3+3.464101615i$$

$$x=-2\sqrt{3}i+3\approx 3-3.464101615i$$

Steps Using the Quadratic Formula

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

$$x^{2}-6x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 21}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 21}}{2}$$

Multiply -4 times 21.

$$x=\frac{-\left(-6\right)±\sqrt{36-84}}{2}$$

Add 36 to -84.

$$x=\frac{-\left(-6\right)±\sqrt{-48}}{2}$$

Take the square root of -48.

$$x=\frac{-\left(-6\right)±4\sqrt{3}i}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±4\sqrt{3}i}{2}$$

Now solve the equation x=\frac{6±4\sqrt{3}i}{2} when ± is plus. Add 6 to 4i\sqrt{3}.

$$x=\frac{6+4\sqrt{3}i}{2}$$

Divide 6+4i\sqrt{3} by 2.

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

Now solve the equation x=\frac{6±4\sqrt{3}i}{2} when ± is minus. Subtract 4i\sqrt{3} from 6.

$$x=\frac{-4\sqrt{3}i+6}{2}$$

Divide 6-4i\sqrt{3} by 2.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$x^{2}-6x+21=0$$

Subtract 21 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-6x=-21$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-21+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-21+9$$

Add -21 to 9.

$$x^{2}-6x+9=-12$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=-12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-12}$$

Simplify.

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

Add 3 to both sides of the equation.

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