Solve x^2+13=5x+6

$$x ^ { 2 } + 13 = 5 x + 6$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{3}i}{2}\approx 2.5+0.866025404i$$

$$x=\frac{-\sqrt{3}i+5}{2}\approx 2.5-0.866025404i$$

Steps Using the Quadratic Formula

Subtract 5x from both sides.

$$x^{2}+13-5x=6$$

Subtract 6 from both sides.

$$x^{2}+13-5x-6=0$$

Subtract 6 from 13 to get 7.

$$x^{2}+7-5x=0$$

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.

$$x^{2}-5x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 7}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-\left(-5\right)±\sqrt{25-28}}{2}$$

Add 25 to -28.

$$x=\frac{-\left(-5\right)±\sqrt{-3}}{2}$$

Take the square root of -3.

$$x=\frac{-\left(-5\right)±\sqrt{3}i}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{3}i}{2}$$

Now solve the equation x=\frac{5±\sqrt{3}i}{2} when ± is plus. Add 5 to i\sqrt{3}.

$$x=\frac{5+\sqrt{3}i}{2}$$

Now solve the equation x=\frac{5±\sqrt{3}i}{2} when ± is minus. Subtract i\sqrt{3} from 5.

$$x=\frac{-\sqrt{3}i+5}{2}$$

The equation is now solved.

$$x=\frac{5+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+5}{2}$$

Steps for Completing the Square

Subtract 5x from both sides.

$$x^{2}+13-5x=6$$

Subtract 13 from both sides.

$$x^{2}-5x=6-13$$

Subtract 13 from 6 to get -7.

$$x^{2}-5x=-7$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-7+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-7+\frac{25}{4}$$

Add -7 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=-\frac{3}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{3}i}{2} x-\frac{5}{2}=-\frac{\sqrt{3}i}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{5+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+5}{2}$$