Solve 9/x+4/x-1=20/x+1

$$\frac { 9 } { x } + \frac { 4 } { x - 1 } = \frac { 20 } { x + 1 }$$

Solve for x

$$x=3$$

$$x=\frac{3}{7}\approx 0.428571429$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+1\right), the least common multiple of x,x-1,x+1.

$$\left(x^{2}-1\right)\times 9+x\left(x+1\right)\times 4=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x^{2}-1 by 9.

$$9x^{2}-9+x\left(x+1\right)\times 4=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x by x+1.

$$9x^{2}-9+\left(x^{2}+x\right)\times 4=x\left(x-1\right)\times 20$$

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

$$9x^{2}-9+4x^{2}+4x=x\left(x-1\right)\times 20$$

Combine 9x^{2} and 4x^{2} to get 13x^{2}.

$$13x^{2}-9+4x=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x by x-1.

$$13x^{2}-9+4x=\left(x^{2}-x\right)\times 20$$

Use the distributive property to multiply x^{2}-x by 20.

$$13x^{2}-9+4x=20x^{2}-20x$$

Subtract 20x^{2} from both sides.

$$13x^{2}-9+4x-20x^{2}=-20x$$

Combine 13x^{2} and -20x^{2} to get -7x^{2}.

$$-7x^{2}-9+4x=-20x$$

Add 20x to both sides.

$$-7x^{2}-9+4x+20x=0$$

Combine 4x and 20x to get 24x.

$$-7x^{2}-9+24x=0$$

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

$$-7x^{2}+24x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=24 ab=-7\left(-9\right)=63$$

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 63.

$$1,63 3,21 7,9$$

Calculate the sum for each pair.

$$1+63=64 3+21=24 7+9=16$$

The solution is the pair that gives sum 24.

$$a=21 b=3$$

Rewrite -7x^{2}+24x-9 as \left(-7x^{2}+21x\right)+\left(3x-9\right).

$$\left(-7x^{2}+21x\right)+\left(3x-9\right)$$

Factor out 7x in the first and -3 in the second group.

$$7x\left(-x+3\right)-3\left(-x+3\right)$$

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

$$\left(-x+3\right)\left(7x-3\right)$$

To find equation solutions, solve -x+3=0 and 7x-3=0.

$$x=3 x=\frac{3}{7}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+1\right), the least common multiple of x,x-1,x+1.

$$\left(x^{2}-1\right)\times 9+x\left(x+1\right)\times 4=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x^{2}-1 by 9.

$$9x^{2}-9+x\left(x+1\right)\times 4=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x by x+1.

$$9x^{2}-9+\left(x^{2}+x\right)\times 4=x\left(x-1\right)\times 20$$

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

$$9x^{2}-9+4x^{2}+4x=x\left(x-1\right)\times 20$$

Combine 9x^{2} and 4x^{2} to get 13x^{2}.

$$13x^{2}-9+4x=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x by x-1.

$$13x^{2}-9+4x=\left(x^{2}-x\right)\times 20$$

Use the distributive property to multiply x^{2}-x by 20.

$$13x^{2}-9+4x=20x^{2}-20x$$

Subtract 20x^{2} from both sides.

$$13x^{2}-9+4x-20x^{2}=-20x$$

Combine 13x^{2} and -20x^{2} to get -7x^{2}.

$$-7x^{2}-9+4x=-20x$$

Add 20x to both sides.

$$-7x^{2}-9+4x+20x=0$$

Combine 4x and 20x to get 24x.

$$-7x^{2}-9+24x=0$$

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.

$$-7x^{2}+24x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 24 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-7\right)\left(-9\right)}}{2\left(-7\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-7\right)\left(-9\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-24±\sqrt{576+28\left(-9\right)}}{2\left(-7\right)}$$

Multiply 28 times -9.

$$x=\frac{-24±\sqrt{576-252}}{2\left(-7\right)}$$

Add 576 to -252.

$$x=\frac{-24±\sqrt{324}}{2\left(-7\right)}$$

Take the square root of 324.

$$x=\frac{-24±18}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-24±18}{-14}$$

Now solve the equation x=\frac{-24±18}{-14} when ± is plus. Add -24 to 18.

$$x=-\frac{6}{-14}$$

Reduce the fraction \frac{-6}{-14} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{7}$$

Now solve the equation x=\frac{-24±18}{-14} when ± is minus. Subtract 18 from -24.

$$x=-\frac{42}{-14}$$

Divide -42 by -14.

$$x=3$$

The equation is now solved.

$$x=\frac{3}{7} x=3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+1\right), the least common multiple of x,x-1,x+1.

$$\left(x^{2}-1\right)\times 9+x\left(x+1\right)\times 4=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x^{2}-1 by 9.

$$9x^{2}-9+x\left(x+1\right)\times 4=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x by x+1.

$$9x^{2}-9+\left(x^{2}+x\right)\times 4=x\left(x-1\right)\times 20$$

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

$$9x^{2}-9+4x^{2}+4x=x\left(x-1\right)\times 20$$

Combine 9x^{2} and 4x^{2} to get 13x^{2}.

$$13x^{2}-9+4x=x\left(x-1\right)\times 20$$

Use the distributive property to multiply x by x-1.

$$13x^{2}-9+4x=\left(x^{2}-x\right)\times 20$$

Use the distributive property to multiply x^{2}-x by 20.

$$13x^{2}-9+4x=20x^{2}-20x$$

Subtract 20x^{2} from both sides.

$$13x^{2}-9+4x-20x^{2}=-20x$$

Combine 13x^{2} and -20x^{2} to get -7x^{2}.

$$-7x^{2}-9+4x=-20x$$

Add 20x to both sides.

$$-7x^{2}-9+4x+20x=0$$

Combine 4x and 20x to get 24x.

$$-7x^{2}-9+24x=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$-7x^{2}+24x=9$$

Divide both sides by -7.

$$\frac{-7x^{2}+24x}{-7}=\frac{9}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{24}{-7}x=\frac{9}{-7}$$

Divide 24 by -7.

$$x^{2}-\frac{24}{7}x=\frac{9}{-7}$$

Divide 9 by -7.

$$x^{2}-\frac{24}{7}x=-\frac{9}{7}$$

Divide -\frac{24}{7}, the coefficient of the x term, by 2 to get -\frac{12}{7}. Then add the square of -\frac{12}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{24}{7}x+\left(-\frac{12}{7}\right)^{2}=-\frac{9}{7}+\left(-\frac{12}{7}\right)^{2}$$

Square -\frac{12}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{24}{7}x+\frac{144}{49}=-\frac{9}{7}+\frac{144}{49}$$

Add -\frac{9}{7} to \frac{144}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{24}{7}x+\frac{144}{49}=\frac{81}{49}$$

Factor x^{2}-\frac{24}{7}x+\frac{144}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{12}{7}\right)^{2}=\frac{81}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{12}{7}\right)^{2}}=\sqrt{\frac{81}{49}}$$

Simplify.

$$x-\frac{12}{7}=\frac{9}{7} x-\frac{12}{7}=-\frac{9}{7}$$

Add \frac{12}{7} to both sides of the equation.

$$x=3 x=\frac{3}{7}$$