Solve 8/n+4/n+10=1

$$\frac { 8 } { n } + \frac { 4 } { n + 10 } = 1$$

Solve for n

$$n=-8$$

$$n=10$$

Steps Using Factoring By Grouping

Variable n cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by n\left(n+10\right), the least common multiple of n,n+10.

$$\left(n+10\right)\times 8+n\times 4=n\left(n+10\right)$$

Use the distributive property to multiply n+10 by 8.

$$8n+80+n\times 4=n\left(n+10\right)$$

Combine 8n and n\times 4 to get 12n.

$$12n+80=n\left(n+10\right)$$

Use the distributive property to multiply n by n+10.

$$12n+80=n^{2}+10n$$

Subtract n^{2} from both sides.

$$12n+80-n^{2}=10n$$

Subtract 10n from both sides.

$$12n+80-n^{2}-10n=0$$

Combine 12n and -10n to get 2n.

$$2n+80-n^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-n^{2}+2n+80=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -n^{2}+an+bn+80. To find a and b, set up a system to be solved.

$$a+b=2 ab=-80=-80$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -80.

$$-1,80 -2,40 -4,20 -5,16 -8,10$$

Calculate the sum for each pair.

$$-1+80=79 -2+40=38 -4+20=16 -5+16=11 -8+10=2$$

The solution is the pair that gives sum 2.

$$a=10 b=-8$$

Rewrite -n^{2}+2n+80 as \left(-n^{2}+10n\right)+\left(-8n+80\right).

$$\left(-n^{2}+10n\right)+\left(-8n+80\right)$$

Factor out -n in the first and -8 in the second group.

$$-n\left(n-10\right)-8\left(n-10\right)$$

Factor out common term n-10 by using distributive property.

$$\left(n-10\right)\left(-n-8\right)$$

To find equation solutions, solve n-10=0 and -n-8=0.

$$n=10 n=-8$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by n\left(n+10\right), the least common multiple of n,n+10.

$$\left(n+10\right)\times 8+n\times 4=n\left(n+10\right)$$

Use the distributive property to multiply n+10 by 8.

$$8n+80+n\times 4=n\left(n+10\right)$$

Combine 8n and n\times 4 to get 12n.

$$12n+80=n\left(n+10\right)$$

Use the distributive property to multiply n by n+10.

$$12n+80=n^{2}+10n$$

Subtract n^{2} from both sides.

$$12n+80-n^{2}=10n$$

Subtract 10n from both sides.

$$12n+80-n^{2}-10n=0$$

Combine 12n and -10n to get 2n.

$$2n+80-n^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-n^{2}+2n+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 80}}{2\left(-1\right)}$$

Square 2.

$$n=\frac{-2±\sqrt{4-4\left(-1\right)\times 80}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-2±\sqrt{4+4\times 80}}{2\left(-1\right)}$$

Multiply 4 times 80.

$$n=\frac{-2±\sqrt{4+320}}{2\left(-1\right)}$$

Add 4 to 320.

$$n=\frac{-2±\sqrt{324}}{2\left(-1\right)}$$

Take the square root of 324.

$$n=\frac{-2±18}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-2±18}{-2}$$

Now solve the equation n=\frac{-2±18}{-2} when ± is plus. Add -2 to 18.

$$n=\frac{16}{-2}$$

Divide 16 by -2.

$$n=-8$$

Now solve the equation n=\frac{-2±18}{-2} when ± is minus. Subtract 18 from -2.

$$n=-\frac{20}{-2}$$

Divide -20 by -2.

$$n=10$$

The equation is now solved.

$$n=-8 n=10$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by n\left(n+10\right), the least common multiple of n,n+10.

$$\left(n+10\right)\times 8+n\times 4=n\left(n+10\right)$$

Use the distributive property to multiply n+10 by 8.

$$8n+80+n\times 4=n\left(n+10\right)$$

Combine 8n and n\times 4 to get 12n.

$$12n+80=n\left(n+10\right)$$

Use the distributive property to multiply n by n+10.

$$12n+80=n^{2}+10n$$

Subtract n^{2} from both sides.

$$12n+80-n^{2}=10n$$

Subtract 10n from both sides.

$$12n+80-n^{2}-10n=0$$

Combine 12n and -10n to get 2n.

$$2n+80-n^{2}=0$$

Subtract 80 from both sides. Anything subtracted from zero gives its negation.

$$2n-n^{2}=-80$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-n^{2}+2n=-80$$

Divide both sides by -1.

$$\frac{-n^{2}+2n}{-1}=-\frac{80}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{2}{-1}n=-\frac{80}{-1}$$

Divide 2 by -1.

$$n^{2}-2n=-\frac{80}{-1}$$

Divide -80 by -1.

$$n^{2}-2n=80$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-2n+1=80+1$$

Add 80 to 1.

$$n^{2}-2n+1=81$$

Factor n^{2}-2n+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-1\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-1\right)^{2}}=\sqrt{81}$$

Simplify.

$$n-1=9 n-1=-9$$

Add 1 to both sides of the equation.

$$n=10 n=-8$$