Solve 4/x+9/x+10=1/2

$$\frac { 4 } { x } + \frac { 9 } { x + 10 } = \frac { 1 } { 2 }$$

Solve for x

$$x=-4$$

$$x=20$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+10\right), the least common multiple of x,x+10,2.

$$\left(2x+20\right)\times 4+2x\times 9=x\left(x+10\right)$$

Use the distributive property to multiply 2x+20 by 4.

$$8x+80+2x\times 9=x\left(x+10\right)$$

Multiply 2 and 9 to get 18.

$$8x+80+18x=x\left(x+10\right)$$

Combine 8x and 18x to get 26x.

$$26x+80=x\left(x+10\right)$$

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

$$26x+80=x^{2}+10x$$

Subtract x^{2} from both sides.

$$26x+80-x^{2}=10x$$

Subtract 10x from both sides.

$$26x+80-x^{2}-10x=0$$

Combine 26x and -10x to get 16x.

$$16x+80-x^{2}=0$$

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

$$-x^{2}+16x+80=0$$

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

$$a+b=16 ab=-80=-80$$

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 16.

$$a=20 b=-4$$

Rewrite -x^{2}+16x+80 as \left(-x^{2}+20x\right)+\left(-4x+80\right).

$$\left(-x^{2}+20x\right)+\left(-4x+80\right)$$

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

$$-x\left(x-20\right)-4\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(-x-4\right)$$

To find equation solutions, solve x-20=0 and -x-4=0.

$$x=20 x=-4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+10\right), the least common multiple of x,x+10,2.

$$\left(2x+20\right)\times 4+2x\times 9=x\left(x+10\right)$$

Use the distributive property to multiply 2x+20 by 4.

$$8x+80+2x\times 9=x\left(x+10\right)$$

Multiply 2 and 9 to get 18.

$$8x+80+18x=x\left(x+10\right)$$

Combine 8x and 18x to get 26x.

$$26x+80=x\left(x+10\right)$$

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

$$26x+80=x^{2}+10x$$

Subtract x^{2} from both sides.

$$26x+80-x^{2}=10x$$

Subtract 10x from both sides.

$$26x+80-x^{2}-10x=0$$

Combine 26x and -10x to get 16x.

$$16x+80-x^{2}=0$$

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.

$$-x^{2}+16x+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 16 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\times 80}}{2\left(-1\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\times 80}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-16±\sqrt{256+4\times 80}}{2\left(-1\right)}$$

Multiply 4 times 80.

$$x=\frac{-16±\sqrt{256+320}}{2\left(-1\right)}$$

Add 256 to 320.

$$x=\frac{-16±\sqrt{576}}{2\left(-1\right)}$$

Take the square root of 576.

$$x=\frac{-16±24}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-16±24}{-2}$$

Now solve the equation x=\frac{-16±24}{-2} when ± is plus. Add -16 to 24.

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

Now solve the equation x=\frac{-16±24}{-2} when ± is minus. Subtract 24 from -16.

$$x=-\frac{40}{-2}$$

Divide -40 by -2.

$$x=20$$

The equation is now solved.

$$x=-4 x=20$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+10\right), the least common multiple of x,x+10,2.

$$\left(2x+20\right)\times 4+2x\times 9=x\left(x+10\right)$$

Use the distributive property to multiply 2x+20 by 4.

$$8x+80+2x\times 9=x\left(x+10\right)$$

Multiply 2 and 9 to get 18.

$$8x+80+18x=x\left(x+10\right)$$

Combine 8x and 18x to get 26x.

$$26x+80=x\left(x+10\right)$$

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

$$26x+80=x^{2}+10x$$

Subtract x^{2} from both sides.

$$26x+80-x^{2}=10x$$

Subtract 10x from both sides.

$$26x+80-x^{2}-10x=0$$

Combine 26x and -10x to get 16x.

$$16x+80-x^{2}=0$$

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

$$16x-x^{2}=-80$$

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.

$$-x^{2}+16x=-80$$

Divide both sides by -1.

$$\frac{-x^{2}+16x}{-1}=-\frac{80}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{16}{-1}x=-\frac{80}{-1}$$

Divide 16 by -1.

$$x^{2}-16x=-\frac{80}{-1}$$

Divide -80 by -1.

$$x^{2}-16x=80$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=80+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=80+64$$

Add 80 to 64.

$$x^{2}-16x+64=144$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{144}$$

Simplify.

$$x-8=12 x-8=-12$$

Add 8 to both sides of the equation.

$$x=20 x=-4$$