Solve 1/24=5(n-2)(n-3)

$$\frac { 1 } { 24 } = 5 ( n - 2 ) ( n - 3 )$$

Solve for n

$$n=\frac{\sqrt{930}}{60}+\frac{5}{2}\approx 3.008265023$$

$$n=-\frac{\sqrt{930}}{60}+\frac{5}{2}\approx 1.991734977$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5 by n-2.

$$\frac{1}{24}=\left(5n-10\right)\left(n-3\right)$$

Use the distributive property to multiply 5n-10 by n-3 and combine like terms.

$$\frac{1}{24}=5n^{2}-25n+30$$

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

$$5n^{2}-25n+30=\frac{1}{24}$$

Subtract \frac{1}{24} from both sides.

$$5n^{2}-25n+30-\frac{1}{24}=0$$

Subtract \frac{1}{24} from 30 to get \frac{719}{24}.

$$5n^{2}-25n+\frac{719}{24}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -25 for b, and \frac{719}{24} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 5\times \frac{719}{24}}}{2\times 5}$$

Square -25.

$$n=\frac{-\left(-25\right)±\sqrt{625-4\times 5\times \frac{719}{24}}}{2\times 5}$$

Multiply -4 times 5.

$$n=\frac{-\left(-25\right)±\sqrt{625-20\times \frac{719}{24}}}{2\times 5}$$

Multiply -20 times \frac{719}{24}.

$$n=\frac{-\left(-25\right)±\sqrt{625-\frac{3595}{6}}}{2\times 5}$$

Add 625 to -\frac{3595}{6}.

$$n=\frac{-\left(-25\right)±\sqrt{\frac{155}{6}}}{2\times 5}$$

Take the square root of \frac{155}{6}.

$$n=\frac{-\left(-25\right)±\frac{\sqrt{930}}{6}}{2\times 5}$$

The opposite of -25 is 25.

$$n=\frac{25±\frac{\sqrt{930}}{6}}{2\times 5}$$

Multiply 2 times 5.

$$n=\frac{25±\frac{\sqrt{930}}{6}}{10}$$

Now solve the equation n=\frac{25±\frac{\sqrt{930}}{6}}{10} when ± is plus. Add 25 to \frac{\sqrt{930}}{6}.

$$n=\frac{\frac{\sqrt{930}}{6}+25}{10}$$

Divide 25+\frac{\sqrt{930}}{6} by 10.

$$n=\frac{\sqrt{930}}{60}+\frac{5}{2}$$

Now solve the equation n=\frac{25±\frac{\sqrt{930}}{6}}{10} when ± is minus. Subtract \frac{\sqrt{930}}{6} from 25.

$$n=\frac{-\frac{\sqrt{930}}{6}+25}{10}$$

Divide 25-\frac{\sqrt{930}}{6} by 10.

$$n=-\frac{\sqrt{930}}{60}+\frac{5}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{930}}{60}+\frac{5}{2} n=-\frac{\sqrt{930}}{60}+\frac{5}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 5 by n-2.

$$\frac{1}{24}=\left(5n-10\right)\left(n-3\right)$$

Use the distributive property to multiply 5n-10 by n-3 and combine like terms.

$$\frac{1}{24}=5n^{2}-25n+30$$

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

$$5n^{2}-25n+30=\frac{1}{24}$$

Subtract 30 from both sides.

$$5n^{2}-25n=\frac{1}{24}-30$$

Subtract 30 from \frac{1}{24} to get -\frac{719}{24}.

$$5n^{2}-25n=-\frac{719}{24}$$

Divide both sides by 5.

$$\frac{5n^{2}-25n}{5}=-\frac{\frac{719}{24}}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$n^{2}+\left(-\frac{25}{5}\right)n=-\frac{\frac{719}{24}}{5}$$

Divide -25 by 5.

$$n^{2}-5n=-\frac{\frac{719}{24}}{5}$$

Divide -\frac{719}{24} by 5.

$$n^{2}-5n=-\frac{719}{120}$$

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.

$$n^{2}-5n+\left(-\frac{5}{2}\right)^{2}=-\frac{719}{120}+\left(-\frac{5}{2}\right)^{2}$$

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

$$n^{2}-5n+\frac{25}{4}=-\frac{719}{120}+\frac{25}{4}$$

Add -\frac{719}{120} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-5n+\frac{25}{4}=\frac{31}{120}$$

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}.

$$\left(n-\frac{5}{2}\right)^{2}=\frac{31}{120}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{2}\right)^{2}}=\sqrt{\frac{31}{120}}$$

Simplify.

$$n-\frac{5}{2}=\frac{\sqrt{930}}{60} n-\frac{5}{2}=-\frac{\sqrt{930}}{60}$$

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

$$n=\frac{\sqrt{930}}{60}+\frac{5}{2} n=-\frac{\sqrt{930}}{60}+\frac{5}{2}$$