Solve x+2/x-x/x+2=24/35

$$\frac { x + 2 } { x } - \frac { x } { x + 2 } = \frac { 24 } { 35 }$$

Solve for x

$$x=5$$

$$x = -\frac{7}{6} = -1\frac{1}{6} \approx -1.166666667$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 35x\left(x+2\right), the least common multiple of x,x+2,35.

$$\left(35x+70\right)\left(x+2\right)-35xx=24x\left(x+2\right)$$

Use the distributive property to multiply 35x+70 by x+2 and combine like terms.

$$35x^{2}+140x+140-35xx=24x\left(x+2\right)$$

Multiply x and x to get x^{2}.

$$35x^{2}+140x+140-35x^{2}=24x\left(x+2\right)$$

Use the distributive property to multiply 24x by x+2.

$$35x^{2}+140x+140-35x^{2}=24x^{2}+48x$$

Subtract 24x^{2} from both sides.

$$35x^{2}+140x+140-35x^{2}-24x^{2}=48x$$

Combine 35x^{2} and -24x^{2} to get 11x^{2}.

$$11x^{2}+140x+140-35x^{2}=48x$$

Subtract 48x from both sides.

$$11x^{2}+140x+140-35x^{2}-48x=0$$

Combine 140x and -48x to get 92x.

$$11x^{2}+92x+140-35x^{2}=0$$

Combine 11x^{2} and -35x^{2} to get -24x^{2}.

$$-24x^{2}+92x+140=0$$

Divide both sides by 4.

$$-6x^{2}+23x+35=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -6x^{2}+ax+bx+35. To find a and b, set up a system to be solved.

$$a+b=23 ab=-6\times 35=-210$$

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 -210.

$$-1,210 -2,105 -3,70 -5,42 -6,35 -7,30 -10,21 -14,15$$

Calculate the sum for each pair.

$$-1+210=209 -2+105=103 -3+70=67 -5+42=37 -6+35=29 -7+30=23 -10+21=11 -14+15=1$$

The solution is the pair that gives sum 23.

$$a=30 b=-7$$

Rewrite -6x^{2}+23x+35 as \left(-6x^{2}+30x\right)+\left(-7x+35\right).

$$\left(-6x^{2}+30x\right)+\left(-7x+35\right)$$

Factor out 6x in the first and 7 in the second group.

$$6x\left(-x+5\right)+7\left(-x+5\right)$$

Factor out common term -x+5 by using distributive property.

$$\left(-x+5\right)\left(6x+7\right)$$

To find equation solutions, solve -x+5=0 and 6x+7=0.

$$x=5 x=-\frac{7}{6}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 35x\left(x+2\right), the least common multiple of x,x+2,35.

$$\left(35x+70\right)\left(x+2\right)-35xx=24x\left(x+2\right)$$

Use the distributive property to multiply 35x+70 by x+2 and combine like terms.

$$35x^{2}+140x+140-35xx=24x\left(x+2\right)$$

Multiply x and x to get x^{2}.

$$35x^{2}+140x+140-35x^{2}=24x\left(x+2\right)$$

Use the distributive property to multiply 24x by x+2.

$$35x^{2}+140x+140-35x^{2}=24x^{2}+48x$$

Subtract 24x^{2} from both sides.

$$35x^{2}+140x+140-35x^{2}-24x^{2}=48x$$

Combine 35x^{2} and -24x^{2} to get 11x^{2}.

$$11x^{2}+140x+140-35x^{2}=48x$$

Subtract 48x from both sides.

$$11x^{2}+140x+140-35x^{2}-48x=0$$

Combine 140x and -48x to get 92x.

$$11x^{2}+92x+140-35x^{2}=0$$

Combine 11x^{2} and -35x^{2} to get -24x^{2}.

$$-24x^{2}+92x+140=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -24 for a, 92 for b, and 140 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-92±\sqrt{92^{2}-4\left(-24\right)\times 140}}{2\left(-24\right)}$$

Square 92.

$$x=\frac{-92±\sqrt{8464-4\left(-24\right)\times 140}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-92±\sqrt{8464+96\times 140}}{2\left(-24\right)}$$

Multiply 96 times 140.

$$x=\frac{-92±\sqrt{8464+13440}}{2\left(-24\right)}$$

Add 8464 to 13440.

$$x=\frac{-92±\sqrt{21904}}{2\left(-24\right)}$$

Take the square root of 21904.

$$x=\frac{-92±148}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{-92±148}{-48}$$

Now solve the equation x=\frac{-92±148}{-48} when ± is plus. Add -92 to 148.

$$x=\frac{56}{-48}$$

Reduce the fraction \frac{56}{-48} to lowest terms by extracting and canceling out 8.

$$x=-\frac{7}{6}$$

Now solve the equation x=\frac{-92±148}{-48} when ± is minus. Subtract 148 from -92.

$$x=-\frac{240}{-48}$$

Divide -240 by -48.

$$x=5$$

The equation is now solved.

$$x=-\frac{7}{6} x=5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 35x\left(x+2\right), the least common multiple of x,x+2,35.

$$\left(35x+70\right)\left(x+2\right)-35xx=24x\left(x+2\right)$$

Use the distributive property to multiply 35x+70 by x+2 and combine like terms.

$$35x^{2}+140x+140-35xx=24x\left(x+2\right)$$

Multiply x and x to get x^{2}.

$$35x^{2}+140x+140-35x^{2}=24x\left(x+2\right)$$

Use the distributive property to multiply 24x by x+2.

$$35x^{2}+140x+140-35x^{2}=24x^{2}+48x$$

Subtract 24x^{2} from both sides.

$$35x^{2}+140x+140-35x^{2}-24x^{2}=48x$$

Combine 35x^{2} and -24x^{2} to get 11x^{2}.

$$11x^{2}+140x+140-35x^{2}=48x$$

Subtract 48x from both sides.

$$11x^{2}+140x+140-35x^{2}-48x=0$$

Combine 140x and -48x to get 92x.

$$11x^{2}+92x+140-35x^{2}=0$$

Subtract 140 from both sides. Anything subtracted from zero gives its negation.

$$11x^{2}+92x-35x^{2}=-140$$

Combine 11x^{2} and -35x^{2} to get -24x^{2}.

$$-24x^{2}+92x=-140$$

Divide both sides by -24.

$$\frac{-24x^{2}+92x}{-24}=-\frac{140}{-24}$$

Dividing by -24 undoes the multiplication by -24.

$$x^{2}+\frac{92}{-24}x=-\frac{140}{-24}$$

Reduce the fraction \frac{92}{-24} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{23}{6}x=-\frac{140}{-24}$$

Reduce the fraction \frac{-140}{-24} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{23}{6}x=\frac{35}{6}$$

Divide -\frac{23}{6}, the coefficient of the x term, by 2 to get -\frac{23}{12}. Then add the square of -\frac{23}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{6}x+\left(-\frac{23}{12}\right)^{2}=\frac{35}{6}+\left(-\frac{23}{12}\right)^{2}$$

Square -\frac{23}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{6}x+\frac{529}{144}=\frac{35}{6}+\frac{529}{144}$$

Add \frac{35}{6} to \frac{529}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{23}{6}x+\frac{529}{144}=\frac{1369}{144}$$

Factor x^{2}-\frac{23}{6}x+\frac{529}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{12}\right)^{2}=\frac{1369}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{12}\right)^{2}}=\sqrt{\frac{1369}{144}}$$

Simplify.

$$x-\frac{23}{12}=\frac{37}{12} x-\frac{23}{12}=-\frac{37}{12}$$

Add \frac{23}{12} to both sides of the equation.

$$x=5 x=-\frac{7}{6}$$