Solve n^2-n+20/left(n+5right)left(n+4right)=4/9

$$\frac{ { n }^{ 2 } -n+20 }{ \left( n+5 \right) \left( n+4 \right) } = \frac{ 4 }{ 9 }$$

Solve for n

$$n=4$$

$$n=5$$

Steps Using Factoring By Grouping

Variable n cannot be equal to any of the values -5,-4 since division by zero is not defined. Multiply both sides of the equation by 9\left(n+4\right)\left(n+5\right), the least common multiple of \left(n+5\right)\left(n+4\right),9.

$$9\left(n^{2}-n+20\right)=4\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 9 by n^{2}-n+20.

$$9n^{2}-9n+180=4\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 4 by n+4.

$$9n^{2}-9n+180=\left(4n+16\right)\left(n+5\right)$$

Use the distributive property to multiply 4n+16 by n+5 and combine like terms.

$$9n^{2}-9n+180=4n^{2}+36n+80$$

Subtract 4n^{2} from both sides.

$$9n^{2}-9n+180-4n^{2}=36n+80$$

Combine 9n^{2} and -4n^{2} to get 5n^{2}.

$$5n^{2}-9n+180=36n+80$$

Subtract 36n from both sides.

$$5n^{2}-9n+180-36n=80$$

Combine -9n and -36n to get -45n.

$$5n^{2}-45n+180=80$$

Subtract 80 from both sides.

$$5n^{2}-45n+180-80=0$$

Subtract 80 from 180 to get 100.

$$5n^{2}-45n+100=0$$

Divide both sides by 5.

$$n^{2}-9n+20=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn+20. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\times 20=20$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 20.

$$-1,-20 -2,-10 -4,-5$$

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -9.

$$a=-5 b=-4$$

Rewrite n^{2}-9n+20 as \left(n^{2}-5n\right)+\left(-4n+20\right).

$$\left(n^{2}-5n\right)+\left(-4n+20\right)$$

Factor out n in the first and -4 in the second group.

$$n\left(n-5\right)-4\left(n-5\right)$$

Factor out common term n-5 by using distributive property.

$$\left(n-5\right)\left(n-4\right)$$

To find equation solutions, solve n-5=0 and n-4=0.

$$n=5 n=4$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -5,-4 since division by zero is not defined. Multiply both sides of the equation by 9\left(n+4\right)\left(n+5\right), the least common multiple of \left(n+5\right)\left(n+4\right),9.

$$9\left(n^{2}-n+20\right)=4\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 9 by n^{2}-n+20.

$$9n^{2}-9n+180=4\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 4 by n+4.

$$9n^{2}-9n+180=\left(4n+16\right)\left(n+5\right)$$

Use the distributive property to multiply 4n+16 by n+5 and combine like terms.

$$9n^{2}-9n+180=4n^{2}+36n+80$$

Subtract 4n^{2} from both sides.

$$9n^{2}-9n+180-4n^{2}=36n+80$$

Combine 9n^{2} and -4n^{2} to get 5n^{2}.

$$5n^{2}-9n+180=36n+80$$

Subtract 36n from both sides.

$$5n^{2}-9n+180-36n=80$$

Combine -9n and -36n to get -45n.

$$5n^{2}-45n+180=80$$

Subtract 80 from both sides.

$$5n^{2}-45n+180-80=0$$

Subtract 80 from 180 to get 100.

$$5n^{2}-45n+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -45 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\times 5\times 100}}{2\times 5}$$

Square -45.

$$n=\frac{-\left(-45\right)±\sqrt{2025-4\times 5\times 100}}{2\times 5}$$

Multiply -4 times 5.

$$n=\frac{-\left(-45\right)±\sqrt{2025-20\times 100}}{2\times 5}$$

Multiply -20 times 100.

$$n=\frac{-\left(-45\right)±\sqrt{2025-2000}}{2\times 5}$$

Add 2025 to -2000.

$$n=\frac{-\left(-45\right)±\sqrt{25}}{2\times 5}$$

Take the square root of 25.

$$n=\frac{-\left(-45\right)±5}{2\times 5}$$

The opposite of -45 is 45.

$$n=\frac{45±5}{2\times 5}$$

Multiply 2 times 5.

$$n=\frac{45±5}{10}$$

Now solve the equation n=\frac{45±5}{10} when ± is plus. Add 45 to 5.

$$n=\frac{50}{10}$$

Divide 50 by 10.

$$n=5$$

Now solve the equation n=\frac{45±5}{10} when ± is minus. Subtract 5 from 45.

$$n=\frac{40}{10}$$

Divide 40 by 10.

$$n=4$$

The equation is now solved.

$$n=5 n=4$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -5,-4 since division by zero is not defined. Multiply both sides of the equation by 9\left(n+4\right)\left(n+5\right), the least common multiple of \left(n+5\right)\left(n+4\right),9.

$$9\left(n^{2}-n+20\right)=4\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 9 by n^{2}-n+20.

$$9n^{2}-9n+180=4\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 4 by n+4.

$$9n^{2}-9n+180=\left(4n+16\right)\left(n+5\right)$$

Use the distributive property to multiply 4n+16 by n+5 and combine like terms.

$$9n^{2}-9n+180=4n^{2}+36n+80$$

Subtract 4n^{2} from both sides.

$$9n^{2}-9n+180-4n^{2}=36n+80$$

Combine 9n^{2} and -4n^{2} to get 5n^{2}.

$$5n^{2}-9n+180=36n+80$$

Subtract 36n from both sides.

$$5n^{2}-9n+180-36n=80$$

Combine -9n and -36n to get -45n.

$$5n^{2}-45n+180=80$$

Subtract 180 from both sides.

$$5n^{2}-45n=80-180$$

Subtract 180 from 80 to get -100.

$$5n^{2}-45n=-100$$

Divide both sides by 5.

$$\frac{5n^{2}-45n}{5}=-\frac{100}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$n^{2}+\left(-\frac{45}{5}\right)n=-\frac{100}{5}$$

Divide -45 by 5.

$$n^{2}-9n=-\frac{100}{5}$$

Divide -100 by 5.

$$n^{2}-9n=-20$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-9n+\left(-\frac{9}{2}\right)^{2}=-20+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-9n+\frac{81}{4}=-20+\frac{81}{4}$$

Add -20 to \frac{81}{4}.

$$n^{2}-9n+\frac{81}{4}=\frac{1}{4}$$

Factor n^{2}-9n+\frac{81}{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}.

$$\left(n-\frac{9}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{9}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$n-\frac{9}{2}=\frac{1}{2} n-\frac{9}{2}=-\frac{1}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$n=5 n=4$$