Solve x/x-5+7x+35/x^2-25=2

$$\frac{ x }{ x-5 } + \frac{ 7x+35 }{ { x }^{ 2 } -25 } = 2$$

Solve for x

$$x=17$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x-5,x^{2}-25.

$$\left(x+5\right)x+7x+35=2\left(x-5\right)\left(x+5\right)$$

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

$$x^{2}+5x+7x+35=2\left(x-5\right)\left(x+5\right)$$

Combine 5x and 7x to get 12x.

$$x^{2}+12x+35=2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x-5.

$$x^{2}+12x+35=\left(2x-10\right)\left(x+5\right)$$

Use the distributive property to multiply 2x-10 by x+5 and combine like terms.

$$x^{2}+12x+35=2x^{2}-50$$

Subtract 2x^{2} from both sides.

$$x^{2}+12x+35-2x^{2}=-50$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+12x+35=-50$$

Add 50 to both sides.

$$-x^{2}+12x+35+50=0$$

Add 35 and 50 to get 85.

$$-x^{2}+12x+85=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+85. To find a and b, set up a system to be solved.

$$a+b=12 ab=-85=-85$$

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 -85.

$$-1,85 -5,17$$

Calculate the sum for each pair.

$$-1+85=84 -5+17=12$$

The solution is the pair that gives sum 12.

$$a=17 b=-5$$

Rewrite -x^{2}+12x+85 as \left(-x^{2}+17x\right)+\left(-5x+85\right).

$$\left(-x^{2}+17x\right)+\left(-5x+85\right)$$

Factor out -x in the first and -5 in the second group.

$$-x\left(x-17\right)-5\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(-x-5\right)$$

To find equation solutions, solve x-17=0 and -x-5=0.

$$x=17 x=-5$$

Variable x cannot be equal to -5.

$$x=17$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x-5,x^{2}-25.

$$\left(x+5\right)x+7x+35=2\left(x-5\right)\left(x+5\right)$$

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

$$x^{2}+5x+7x+35=2\left(x-5\right)\left(x+5\right)$$

Combine 5x and 7x to get 12x.

$$x^{2}+12x+35=2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x-5.

$$x^{2}+12x+35=\left(2x-10\right)\left(x+5\right)$$

Use the distributive property to multiply 2x-10 by x+5 and combine like terms.

$$x^{2}+12x+35=2x^{2}-50$$

Subtract 2x^{2} from both sides.

$$x^{2}+12x+35-2x^{2}=-50$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+12x+35=-50$$

Add 50 to both sides.

$$-x^{2}+12x+35+50=0$$

Add 35 and 50 to get 85.

$$-x^{2}+12x+85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12 for b, and 85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-1\right)\times 85}}{2\left(-1\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-1\right)\times 85}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12±\sqrt{144+4\times 85}}{2\left(-1\right)}$$

Multiply 4 times 85.

$$x=\frac{-12±\sqrt{144+340}}{2\left(-1\right)}$$

Add 144 to 340.

$$x=\frac{-12±\sqrt{484}}{2\left(-1\right)}$$

Take the square root of 484.

$$x=\frac{-12±22}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±22}{-2}$$

Now solve the equation x=\frac{-12±22}{-2} when ± is plus. Add -12 to 22.

$$x=\frac{10}{-2}$$

Divide 10 by -2.

$$x=-5$$

Now solve the equation x=\frac{-12±22}{-2} when ± is minus. Subtract 22 from -12.

$$x=-\frac{34}{-2}$$

Divide -34 by -2.

$$x=17$$

The equation is now solved.

$$x=-5 x=17$$

Variable x cannot be equal to -5.

$$x=17$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x-5,x^{2}-25.

$$\left(x+5\right)x+7x+35=2\left(x-5\right)\left(x+5\right)$$

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

$$x^{2}+5x+7x+35=2\left(x-5\right)\left(x+5\right)$$

Combine 5x and 7x to get 12x.

$$x^{2}+12x+35=2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x-5.

$$x^{2}+12x+35=\left(2x-10\right)\left(x+5\right)$$

Use the distributive property to multiply 2x-10 by x+5 and combine like terms.

$$x^{2}+12x+35=2x^{2}-50$$

Subtract 2x^{2} from both sides.

$$x^{2}+12x+35-2x^{2}=-50$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+12x+35=-50$$

Subtract 35 from both sides.

$$-x^{2}+12x=-50-35$$

Subtract 35 from -50 to get -85.

$$-x^{2}+12x=-85$$

Divide both sides by -1.

$$\frac{-x^{2}+12x}{-1}=-\frac{85}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{12}{-1}x=-\frac{85}{-1}$$

Divide 12 by -1.

$$x^{2}-12x=-\frac{85}{-1}$$

Divide -85 by -1.

$$x^{2}-12x=85$$

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

$$x^{2}-12x+\left(-6\right)^{2}=85+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=85+36$$

Add 85 to 36.

$$x^{2}-12x+36=121$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=121$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{121}$$

Simplify.

$$x-6=11 x-6=-11$$

Add 6 to both sides of the equation.

$$x=17 x=-5$$

Variable x cannot be equal to -5.

$$x=17$$