Solve 3/x+5/x-2=30/x+2

$$\frac { 3 } { x } + \frac { 5 } { x - 2 } = \frac { 30 } { x + 2 }$$

Solve for x

$$x=3$$

$$x=\frac{2}{11}\approx 0.181818182$$

Steps Using Factoring By Grouping

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

$$\left(x^{2}-4\right)\times 3+x\left(x+2\right)\times 5=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x^{2}-4 by 3.

$$3x^{2}-12+x\left(x+2\right)\times 5=x\left(x-2\right)\times 30$$

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

$$3x^{2}-12+\left(x^{2}+2x\right)\times 5=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x^{2}+2x by 5.

$$3x^{2}-12+5x^{2}+10x=x\left(x-2\right)\times 30$$

Combine 3x^{2} and 5x^{2} to get 8x^{2}.

$$8x^{2}-12+10x=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x by x-2.

$$8x^{2}-12+10x=\left(x^{2}-2x\right)\times 30$$

Use the distributive property to multiply x^{2}-2x by 30.

$$8x^{2}-12+10x=30x^{2}-60x$$

Subtract 30x^{2} from both sides.

$$8x^{2}-12+10x-30x^{2}=-60x$$

Combine 8x^{2} and -30x^{2} to get -22x^{2}.

$$-22x^{2}-12+10x=-60x$$

Add 60x to both sides.

$$-22x^{2}-12+10x+60x=0$$

Combine 10x and 60x to get 70x.

$$-22x^{2}-12+70x=0$$

Divide both sides by 2.

$$-11x^{2}-6+35x=0$$

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

$$-11x^{2}+35x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -11x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=35 ab=-11\left(-6\right)=66$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 66.

$$1,66 2,33 3,22 6,11$$

Calculate the sum for each pair.

$$1+66=67 2+33=35 3+22=25 6+11=17$$

The solution is the pair that gives sum 35.

$$a=33 b=2$$

Rewrite -11x^{2}+35x-6 as \left(-11x^{2}+33x\right)+\left(2x-6\right).

$$\left(-11x^{2}+33x\right)+\left(2x-6\right)$$

Factor out 11x in the first and -2 in the second group.

$$11x\left(-x+3\right)-2\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(11x-2\right)$$

To find equation solutions, solve -x+3=0 and 11x-2=0.

$$x=3 x=\frac{2}{11}$$

Steps Using the Quadratic Formula

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

$$\left(x^{2}-4\right)\times 3+x\left(x+2\right)\times 5=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x^{2}-4 by 3.

$$3x^{2}-12+x\left(x+2\right)\times 5=x\left(x-2\right)\times 30$$

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

$$3x^{2}-12+\left(x^{2}+2x\right)\times 5=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x^{2}+2x by 5.

$$3x^{2}-12+5x^{2}+10x=x\left(x-2\right)\times 30$$

Combine 3x^{2} and 5x^{2} to get 8x^{2}.

$$8x^{2}-12+10x=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x by x-2.

$$8x^{2}-12+10x=\left(x^{2}-2x\right)\times 30$$

Use the distributive property to multiply x^{2}-2x by 30.

$$8x^{2}-12+10x=30x^{2}-60x$$

Subtract 30x^{2} from both sides.

$$8x^{2}-12+10x-30x^{2}=-60x$$

Combine 8x^{2} and -30x^{2} to get -22x^{2}.

$$-22x^{2}-12+10x=-60x$$

Add 60x to both sides.

$$-22x^{2}-12+10x+60x=0$$

Combine 10x and 60x to get 70x.

$$-22x^{2}-12+70x=0$$

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.

$$-22x^{2}+70x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -22 for a, 70 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}-4\left(-22\right)\left(-12\right)}}{2\left(-22\right)}$$

Square 70.

$$x=\frac{-70±\sqrt{4900-4\left(-22\right)\left(-12\right)}}{2\left(-22\right)}$$

Multiply -4 times -22.

$$x=\frac{-70±\sqrt{4900+88\left(-12\right)}}{2\left(-22\right)}$$

Multiply 88 times -12.

$$x=\frac{-70±\sqrt{4900-1056}}{2\left(-22\right)}$$

Add 4900 to -1056.

$$x=\frac{-70±\sqrt{3844}}{2\left(-22\right)}$$

Take the square root of 3844.

$$x=\frac{-70±62}{2\left(-22\right)}$$

Multiply 2 times -22.

$$x=\frac{-70±62}{-44}$$

Now solve the equation x=\frac{-70±62}{-44} when ± is plus. Add -70 to 62.

$$x=-\frac{8}{-44}$$

Reduce the fraction \frac{-8}{-44} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{11}$$

Now solve the equation x=\frac{-70±62}{-44} when ± is minus. Subtract 62 from -70.

$$x=-\frac{132}{-44}$$

Divide -132 by -44.

$$x=3$$

The equation is now solved.

$$x=\frac{2}{11} x=3$$

Steps for Completing the Square

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

$$\left(x^{2}-4\right)\times 3+x\left(x+2\right)\times 5=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x^{2}-4 by 3.

$$3x^{2}-12+x\left(x+2\right)\times 5=x\left(x-2\right)\times 30$$

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

$$3x^{2}-12+\left(x^{2}+2x\right)\times 5=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x^{2}+2x by 5.

$$3x^{2}-12+5x^{2}+10x=x\left(x-2\right)\times 30$$

Combine 3x^{2} and 5x^{2} to get 8x^{2}.

$$8x^{2}-12+10x=x\left(x-2\right)\times 30$$

Use the distributive property to multiply x by x-2.

$$8x^{2}-12+10x=\left(x^{2}-2x\right)\times 30$$

Use the distributive property to multiply x^{2}-2x by 30.

$$8x^{2}-12+10x=30x^{2}-60x$$

Subtract 30x^{2} from both sides.

$$8x^{2}-12+10x-30x^{2}=-60x$$

Combine 8x^{2} and -30x^{2} to get -22x^{2}.

$$-22x^{2}-12+10x=-60x$$

Add 60x to both sides.

$$-22x^{2}-12+10x+60x=0$$

Combine 10x and 60x to get 70x.

$$-22x^{2}-12+70x=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$-22x^{2}+70x=12$$

Divide both sides by -22.

$$\frac{-22x^{2}+70x}{-22}=\frac{12}{-22}$$

Dividing by -22 undoes the multiplication by -22.

$$x^{2}+\frac{70}{-22}x=\frac{12}{-22}$$

Reduce the fraction \frac{70}{-22} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{35}{11}x=\frac{12}{-22}$$

Reduce the fraction \frac{12}{-22} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{35}{11}x=-\frac{6}{11}$$

Divide -\frac{35}{11}, the coefficient of the x term, by 2 to get -\frac{35}{22}. Then add the square of -\frac{35}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{11}x+\left(-\frac{35}{22}\right)^{2}=-\frac{6}{11}+\left(-\frac{35}{22}\right)^{2}$$

Square -\frac{35}{22} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{11}x+\frac{1225}{484}=-\frac{6}{11}+\frac{1225}{484}$$

Add -\frac{6}{11} to \frac{1225}{484} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{35}{11}x+\frac{1225}{484}=\frac{961}{484}$$

Factor x^{2}-\frac{35}{11}x+\frac{1225}{484}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{22}\right)^{2}=\frac{961}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{22}\right)^{2}}=\sqrt{\frac{961}{484}}$$

Simplify.

$$x-\frac{35}{22}=\frac{31}{22} x-\frac{35}{22}=-\frac{31}{22}$$

Add \frac{35}{22} to both sides of the equation.

$$x=3 x=\frac{2}{11}$$