Solve 9/x-9+x/x-1=-5x-13/x^2-10x+9

$$\frac{ 9 }{ x-9 } + \frac{ x }{ x-1 } = \frac{ -5x-13 }{ { x }^{ 2 } -10x+9 }$$

Solve for x

$$x=-4$$

$$x=-1$$

Steps Using Factoring

Variable x cannot be equal to any of the values 1,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x-1\right), the least common multiple of x-9,x-1,x^{2}-10x+9.

$$\left(x-1\right)\times 9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-1 by 9.

$$9x-9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-9 by x.

$$9x-9+x^{2}-9x=-5x-13$$

Combine 9x and -9x to get 0.

$$-9+x^{2}=-5x-13$$

Add 5x to both sides.

$$-9+x^{2}+5x=-13$$

Add 13 to both sides.

$$-9+x^{2}+5x+13=0$$

Add -9 and 13 to get 4.

$$4+x^{2}+5x=0$$

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

$$x^{2}+5x+4=0$$

To solve the equation, factor x^{2}+5x+4 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=5 ab=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=1 b=4$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

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

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

$$x=-1 x=-4$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 1,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x-1\right), the least common multiple of x-9,x-1,x^{2}-10x+9.

$$\left(x-1\right)\times 9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-1 by 9.

$$9x-9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-9 by x.

$$9x-9+x^{2}-9x=-5x-13$$

Combine 9x and -9x to get 0.

$$-9+x^{2}=-5x-13$$

Add 5x to both sides.

$$-9+x^{2}+5x=-13$$

Add 13 to both sides.

$$-9+x^{2}+5x+13=0$$

Add -9 and 13 to get 4.

$$4+x^{2}+5x=0$$

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

$$x^{2}+5x+4=0$$

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

$$a+b=5 ab=1\times 4=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=1 b=4$$

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

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

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

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

Factor out common term x+1 by using distributive property.

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

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

$$x=-1 x=-4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x-1\right), the least common multiple of x-9,x-1,x^{2}-10x+9.

$$\left(x-1\right)\times 9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-1 by 9.

$$9x-9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-9 by x.

$$9x-9+x^{2}-9x=-5x-13$$

Combine 9x and -9x to get 0.

$$-9+x^{2}=-5x-13$$

Add 5x to both sides.

$$-9+x^{2}+5x=-13$$

Add 13 to both sides.

$$-9+x^{2}+5x+13=0$$

Add -9 and 13 to get 4.

$$4+x^{2}+5x=0$$

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.

$$x^{2}+5x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 4}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 4}}{2}$$

Multiply -4 times 4.

$$x=\frac{-5±\sqrt{25-16}}{2}$$

Add 25 to -16.

$$x=\frac{-5±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-5±3}{2}$$

Now solve the equation x=\frac{-5±3}{2} when ± is plus. Add -5 to 3.

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

Divide -2 by 2.

$$x=-1$$

Now solve the equation x=\frac{-5±3}{2} when ± is minus. Subtract 3 from -5.

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

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=-1 x=-4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x-1\right), the least common multiple of x-9,x-1,x^{2}-10x+9.

$$\left(x-1\right)\times 9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-1 by 9.

$$9x-9+\left(x-9\right)x=-5x-13$$

Use the distributive property to multiply x-9 by x.

$$9x-9+x^{2}-9x=-5x-13$$

Combine 9x and -9x to get 0.

$$-9+x^{2}=-5x-13$$

Add 5x to both sides.

$$-9+x^{2}+5x=-13$$

Add 9 to both sides.

$$x^{2}+5x=-13+9$$

Add -13 and 9 to get -4.

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

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

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

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

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

Add -4 to \frac{25}{4}.

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

Factor x^{2}+5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract \frac{5}{2} from both sides of the equation.

$$x=-1 x=-4$$