Solve 0=n^2-5n+24

$$0 = n ^ { 2 } - 5 n + 24$$

Solve for n

$$n=\frac{5+\sqrt{71}i}{2}\approx 2.5+4.213074887i$$

$$n=\frac{-\sqrt{71}i+5}{2}\approx 2.5-4.213074887i$$

Steps Using the Quadratic Formula

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

$$n^{2}-5n+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 24}}{2}$$

Square -5.

$$n=\frac{-\left(-5\right)±\sqrt{25-4\times 24}}{2}$$

Multiply -4 times 24.

$$n=\frac{-\left(-5\right)±\sqrt{25-96}}{2}$$

Add 25 to -96.

$$n=\frac{-\left(-5\right)±\sqrt{-71}}{2}$$

Take the square root of -71.

$$n=\frac{-\left(-5\right)±\sqrt{71}i}{2}$$

The opposite of -5 is 5.

$$n=\frac{5±\sqrt{71}i}{2}$$

Now solve the equation n=\frac{5±\sqrt{71}i}{2} when ± is plus. Add 5 to i\sqrt{71}.

$$n=\frac{5+\sqrt{71}i}{2}$$

Now solve the equation n=\frac{5±\sqrt{71}i}{2} when ± is minus. Subtract i\sqrt{71} from 5.

$$n=\frac{-\sqrt{71}i+5}{2}$$

The equation is now solved.

$$n=\frac{5+\sqrt{71}i}{2} n=\frac{-\sqrt{71}i+5}{2}$$

Steps for Completing the Square

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

$$n^{2}-5n+24=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$n^{2}-5n=-24$$

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.

$$n^{2}-5n+\left(-\frac{5}{2}\right)^{2}=-24+\left(-\frac{5}{2}\right)^{2}$$

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

$$n^{2}-5n+\frac{25}{4}=-24+\frac{25}{4}$$

Add -24 to \frac{25}{4}.

$$n^{2}-5n+\frac{25}{4}=-\frac{71}{4}$$

Factor n^{2}-5n+\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}.

$$\left(n-\frac{5}{2}\right)^{2}=-\frac{71}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{71}{4}}$$

Simplify.

$$n-\frac{5}{2}=\frac{\sqrt{71}i}{2} n-\frac{5}{2}=-\frac{\sqrt{71}i}{2}$$

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

$$n=\frac{5+\sqrt{71}i}{2} n=\frac{-\sqrt{71}i+5}{2}$$