Solve 9x/x+1-9/x^2+x=5

$$\frac { 9 x } { x + 1 } - \frac { 9 } { x ^ { 2 } + x } = 5$$

Solve for x

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x+1,x^{2}+x.

$$x\times 9x-9=5x\left(x+1\right)$$

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

$$x^{2}\times 9-9=5x\left(x+1\right)$$

Use the distributive property to multiply 5x by x+1.

$$x^{2}\times 9-9=5x^{2}+5x$$

Subtract 5x^{2} from both sides.

$$x^{2}\times 9-9-5x^{2}=5x$$

Combine x^{2}\times 9 and -5x^{2} to get 4x^{2}.

$$4x^{2}-9=5x$$

Subtract 5x from both sides.

$$4x^{2}-9-5x=0$$

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

$$4x^{2}-5x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=-5 ab=4\left(-9\right)=-36$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -5.

$$a=-9 b=4$$

Rewrite 4x^{2}-5x-9 as \left(4x^{2}-9x\right)+\left(4x-9\right).

$$\left(4x^{2}-9x\right)+\left(4x-9\right)$$

Factor out x in 4x^{2}-9x.

$$x\left(4x-9\right)+4x-9$$

Factor out common term 4x-9 by using distributive property.

$$\left(4x-9\right)\left(x+1\right)$$

To find equation solutions, solve 4x-9=0 and x+1=0.

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

Variable x cannot be equal to -1.

$$x=\frac{9}{4}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x+1,x^{2}+x.

$$x\times 9x-9=5x\left(x+1\right)$$

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

$$x^{2}\times 9-9=5x\left(x+1\right)$$

Use the distributive property to multiply 5x by x+1.

$$x^{2}\times 9-9=5x^{2}+5x$$

Subtract 5x^{2} from both sides.

$$x^{2}\times 9-9-5x^{2}=5x$$

Combine x^{2}\times 9 and -5x^{2} to get 4x^{2}.

$$4x^{2}-9=5x$$

Subtract 5x from both sides.

$$4x^{2}-9-5x=0$$

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.

$$4x^{2}-5x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -5 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 4\left(-9\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-5\right)±\sqrt{25-16\left(-9\right)}}{2\times 4}$$

Multiply -16 times -9.

$$x=\frac{-\left(-5\right)±\sqrt{25+144}}{2\times 4}$$

Add 25 to 144.

$$x=\frac{-\left(-5\right)±\sqrt{169}}{2\times 4}$$

Take the square root of 169.

$$x=\frac{-\left(-5\right)±13}{2\times 4}$$

The opposite of -5 is 5.

$$x=\frac{5±13}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{5±13}{8}$$

Now solve the equation x=\frac{5±13}{8} when ± is plus. Add 5 to 13.

$$x=\frac{18}{8}$$

Reduce the fraction \frac{18}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{4}$$

Now solve the equation x=\frac{5±13}{8} when ± is minus. Subtract 13 from 5.

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

Divide -8 by 8.

$$x=-1$$

The equation is now solved.

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

Variable x cannot be equal to -1.

$$x=\frac{9}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x+1,x^{2}+x.

$$x\times 9x-9=5x\left(x+1\right)$$

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

$$x^{2}\times 9-9=5x\left(x+1\right)$$

Use the distributive property to multiply 5x by x+1.

$$x^{2}\times 9-9=5x^{2}+5x$$

Subtract 5x^{2} from both sides.

$$x^{2}\times 9-9-5x^{2}=5x$$

Combine x^{2}\times 9 and -5x^{2} to get 4x^{2}.

$$4x^{2}-9=5x$$

Subtract 5x from both sides.

$$4x^{2}-9-5x=0$$

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

$$4x^{2}-5x=9$$

Divide both sides by 4.

$$\frac{4x^{2}-5x}{4}=\frac{9}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{5}{4}x=\frac{9}{4}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=\frac{9}{4}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{9}{4}+\frac{25}{64}$$

Add \frac{9}{4} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{169}{64}$$

Factor x^{2}-\frac{5}{4}x+\frac{25}{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}.

$$\left(x-\frac{5}{8}\right)^{2}=\frac{169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Simplify.

$$x-\frac{5}{8}=\frac{13}{8} x-\frac{5}{8}=-\frac{13}{8}$$

Add \frac{5}{8} to both sides of the equation.

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

Variable x cannot be equal to -1.

$$x=\frac{9}{4}$$