Solve x^2+13=6x

$$x ^ { 2 } + 13 = 6 x$$

Solve for x (complex solution)

$$x=3+2i$$

$$x=3-2i$$

Steps Using the Quadratic Formula

Subtract 6x from both sides.

$$x^{2}+13-6x=0$$

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.

$$x^{2}-6x+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 13}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 13}}{2}$$

Multiply -4 times 13.

$$x=\frac{-\left(-6\right)±\sqrt{36-52}}{2}$$

Add 36 to -52.

$$x=\frac{-\left(-6\right)±\sqrt{-16}}{2}$$

Take the square root of -16.

$$x=\frac{-\left(-6\right)±4i}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±4i}{2}$$

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

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

Divide 6+4i by 2.

$$x=3+2i$$

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

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

Divide 6-4i by 2.

$$x=3-2i$$

The equation is now solved.

$$x=3+2i x=3-2i$$

Steps for Completing the Square

Subtract 6x from both sides.

$$x^{2}+13-6x=0$$

Subtract 13 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-6x=-13$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=-13+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-13+9$$

Add -13 to 9.

$$x^{2}-6x+9=-4$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-3=2i x-3=-2i$$

Add 3 to both sides of the equation.

$$x=3+2i x=3-2i$$