Solve x^2=6x-18

$$x ^ { 2 } = 6 x - 18$$

Solve for x (complex solution)

$$x=3+3i$$

$$x=3-3i$$

Steps Using the Quadratic Formula

Subtract 6x from both sides.

$$x^{2}-6x=-18$$

Add 18 to both sides.

$$x^{2}-6x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 18}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-\left(-6\right)±\sqrt{36-72}}{2}$$

Add 36 to -72.

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

Take the square root of -36.

$$x=\frac{-\left(-6\right)±6i}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±6i}{2}$$

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

$$x=\frac{6+6i}{2}$$

Divide 6+6i by 2.

$$x=3+3i$$

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

$$x=\frac{6-6i}{2}$$

Divide 6-6i by 2.

$$x=3-3i$$

The equation is now solved.

$$x=3+3i x=3-3i$$

Steps for Completing the Square

Subtract 6x from both sides.

$$x^{2}-6x=-18$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=-18+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-18+9$$

Add -18 to 9.

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

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}.

$$\left(x-3\right)^{2}=-9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-9}$$

Simplify.

$$x-3=3i x-3=-3i$$

Add 3 to both sides of the equation.

$$x=3+3i x=3-3i$$