Solve n(n-1)/(n+5)(n+4)=2/25

$$\frac { n ( n - 1 ) } { ( n + 5 ) ( n + 4 ) } = \frac { 2 } { 25 }$$

Solve for n

$$n = \frac{\sqrt{5529} + 43}{46} \approx 2.551244473$$

$$n=\frac{43-\sqrt{5529}}{46}\approx -0.681679256$$

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 25\left(n+4\right)\left(n+5\right), the least common multiple of \left(n+5\right)\left(n+4\right),25.

$$25n\left(n-1\right)=2\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 25n by n-1.

$$25n^{2}-25n=2\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 2 by n+4.

$$25n^{2}-25n=\left(2n+8\right)\left(n+5\right)$$

Use the distributive property to multiply 2n+8 by n+5 and combine like terms.

$$25n^{2}-25n=2n^{2}+18n+40$$

Subtract 2n^{2} from both sides.

$$25n^{2}-25n-2n^{2}=18n+40$$

Combine 25n^{2} and -2n^{2} to get 23n^{2}.

$$23n^{2}-25n=18n+40$$

Subtract 18n from both sides.

$$23n^{2}-25n-18n=40$$

Combine -25n and -18n to get -43n.

$$23n^{2}-43n=40$$

Subtract 40 from both sides.

$$23n^{2}-43n-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 23 for a, -43 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-43\right)±\sqrt{\left(-43\right)^{2}-4\times 23\left(-40\right)}}{2\times 23}$$

Square -43.

$$n=\frac{-\left(-43\right)±\sqrt{1849-4\times 23\left(-40\right)}}{2\times 23}$$

Multiply -4 times 23.

$$n=\frac{-\left(-43\right)±\sqrt{1849-92\left(-40\right)}}{2\times 23}$$

Multiply -92 times -40.

$$n=\frac{-\left(-43\right)±\sqrt{1849+3680}}{2\times 23}$$

Add 1849 to 3680.

$$n=\frac{-\left(-43\right)±\sqrt{5529}}{2\times 23}$$

The opposite of -43 is 43.

$$n=\frac{43±\sqrt{5529}}{2\times 23}$$

Multiply 2 times 23.

$$n=\frac{43±\sqrt{5529}}{46}$$

Now solve the equation n=\frac{43±\sqrt{5529}}{46} when ± is plus. Add 43 to \sqrt{5529}.

$$n=\frac{\sqrt{5529}+43}{46}$$

Now solve the equation n=\frac{43±\sqrt{5529}}{46} when ± is minus. Subtract \sqrt{5529} from 43.

$$n=\frac{43-\sqrt{5529}}{46}$$

The equation is now solved.

$$n=\frac{\sqrt{5529}+43}{46} n=\frac{43-\sqrt{5529}}{46}$$

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 25\left(n+4\right)\left(n+5\right), the least common multiple of \left(n+5\right)\left(n+4\right),25.

$$25n\left(n-1\right)=2\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 25n by n-1.

$$25n^{2}-25n=2\left(n+4\right)\left(n+5\right)$$

Use the distributive property to multiply 2 by n+4.

$$25n^{2}-25n=\left(2n+8\right)\left(n+5\right)$$

Use the distributive property to multiply 2n+8 by n+5 and combine like terms.

$$25n^{2}-25n=2n^{2}+18n+40$$

Subtract 2n^{2} from both sides.

$$25n^{2}-25n-2n^{2}=18n+40$$

Combine 25n^{2} and -2n^{2} to get 23n^{2}.

$$23n^{2}-25n=18n+40$$

Subtract 18n from both sides.

$$23n^{2}-25n-18n=40$$

Combine -25n and -18n to get -43n.

$$23n^{2}-43n=40$$

Divide both sides by 23.

$$\frac{23n^{2}-43n}{23}=\frac{40}{23}$$

Dividing by 23 undoes the multiplication by 23.

$$n^{2}-\frac{43}{23}n=\frac{40}{23}$$

Divide -\frac{43}{23}, the coefficient of the x term, by 2 to get -\frac{43}{46}. Then add the square of -\frac{43}{46} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{43}{23}n+\left(-\frac{43}{46}\right)^{2}=\frac{40}{23}+\left(-\frac{43}{46}\right)^{2}$$

Square -\frac{43}{46} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{43}{23}n+\frac{1849}{2116}=\frac{40}{23}+\frac{1849}{2116}$$

Add \frac{40}{23} to \frac{1849}{2116} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{43}{23}n+\frac{1849}{2116}=\frac{5529}{2116}$$

Factor n^{2}-\frac{43}{23}n+\frac{1849}{2116}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{43}{46}\right)^{2}=\frac{5529}{2116}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{43}{46}\right)^{2}}=\sqrt{\frac{5529}{2116}}$$

Simplify.

$$n-\frac{43}{46}=\frac{\sqrt{5529}}{46} n-\frac{43}{46}=-\frac{\sqrt{5529}}{46}$$

Add \frac{43}{46} to both sides of the equation.

$$n=\frac{\sqrt{5529}+43}{46} n=\frac{43-\sqrt{5529}}{46}$$