Solve (n-2)(n-3)=224

$$( n - 2 ) ( n - 3 ) = 224$$

Solve for n

$$n = \frac{\sqrt{897} + 5}{2} \approx 17.474979132$$

$$n=\frac{5-\sqrt{897}}{2}\approx -12.474979132$$

Steps Using the Quadratic Formula

Use the distributive property to multiply n-2 by n-3 and combine like terms.

$$n^{2}-5n+6=224$$

Subtract 224 from both sides.

$$n^{2}-5n+6-224=0$$

Subtract 224 from 6 to get -218.

$$n^{2}-5n-218=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -218 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-218\right)}}{2}$$

Square -5.

$$n=\frac{-\left(-5\right)±\sqrt{25-4\left(-218\right)}}{2}$$

Multiply -4 times -218.

$$n=\frac{-\left(-5\right)±\sqrt{25+872}}{2}$$

Add 25 to 872.

$$n=\frac{-\left(-5\right)±\sqrt{897}}{2}$$

The opposite of -5 is 5.

$$n=\frac{5±\sqrt{897}}{2}$$

Now solve the equation n=\frac{5±\sqrt{897}}{2} when ± is plus. Add 5 to \sqrt{897}.

$$n=\frac{\sqrt{897}+5}{2}$$

Now solve the equation n=\frac{5±\sqrt{897}}{2} when ± is minus. Subtract \sqrt{897} from 5.

$$n=\frac{5-\sqrt{897}}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{897}+5}{2} n=\frac{5-\sqrt{897}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply n-2 by n-3 and combine like terms.

$$n^{2}-5n+6=224$$

Subtract 6 from both sides.

$$n^{2}-5n=224-6$$

Subtract 6 from 224 to get 218.

$$n^{2}-5n=218$$

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.

$$n^{2}-5n+\left(-\frac{5}{2}\right)^{2}=218+\left(-\frac{5}{2}\right)^{2}$$

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

$$n^{2}-5n+\frac{25}{4}=218+\frac{25}{4}$$

Add 218 to \frac{25}{4}.

$$n^{2}-5n+\frac{25}{4}=\frac{897}{4}$$

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}.

$$\left(n-\frac{5}{2}\right)^{2}=\frac{897}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{2}\right)^{2}}=\sqrt{\frac{897}{4}}$$

Simplify.

$$n-\frac{5}{2}=\frac{\sqrt{897}}{2} n-\frac{5}{2}=-\frac{\sqrt{897}}{2}$$

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

$$n=\frac{\sqrt{897}+5}{2} n=\frac{5-\sqrt{897}}{2}$$