Solve x+6/x+5+10/x^2-25=2

$$\frac { x + 6 } { x + 5 } + \frac { 10 } { x ^ { 2 } - 25 } = 2$$

Solve for x

$$x=6$$

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.

$$\left(x-5\right)\left(x+6\right)+10=2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x-5 by x+6 and combine like terms.

$$x^{2}+x-30+10=2\left(x-5\right)\left(x+5\right)$$

Add -30 and 10 to get -20.

$$x^{2}+x-20=2\left(x-5\right)\left(x+5\right)$$

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

$$x^{2}+x-20=\left(2x-10\right)\left(x+5\right)$$

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

$$x^{2}+x-20=2x^{2}-50$$

Subtract 2x^{2} from both sides.

$$x^{2}+x-20-2x^{2}=-50$$

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

$$-x^{2}+x-20=-50$$

Add 50 to both sides.

$$-x^{2}+x-20+50=0$$

Add -20 and 50 to get 30.

$$-x^{2}+x+30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+30. To find a and b, set up a system to be solved.

$$a+b=1 ab=-30=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Calculate the sum for each pair.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

The solution is the pair that gives sum 1.

$$a=6 b=-5$$

Rewrite -x^{2}+x+30 as \left(-x^{2}+6x\right)+\left(-5x+30\right).

$$\left(-x^{2}+6x\right)+\left(-5x+30\right)$$

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

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

Factor out common term x-6 by using distributive property.

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

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

$$x=6 x=-5$$

Variable x cannot be equal to -5.

$$x=6$$

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.

$$\left(x-5\right)\left(x+6\right)+10=2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x-5 by x+6 and combine like terms.

$$x^{2}+x-30+10=2\left(x-5\right)\left(x+5\right)$$

Add -30 and 10 to get -20.

$$x^{2}+x-20=2\left(x-5\right)\left(x+5\right)$$

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

$$x^{2}+x-20=\left(2x-10\right)\left(x+5\right)$$

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

$$x^{2}+x-20=2x^{2}-50$$

Subtract 2x^{2} from both sides.

$$x^{2}+x-20-2x^{2}=-50$$

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

$$-x^{2}+x-20=-50$$

Add 50 to both sides.

$$-x^{2}+x-20+50=0$$

Add -20 and 50 to get 30.

$$-x^{2}+x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1±\sqrt{1+4\times 30}}{2\left(-1\right)}$$

Multiply 4 times 30.

$$x=\frac{-1±\sqrt{1+120}}{2\left(-1\right)}$$

Add 1 to 120.

$$x=\frac{-1±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-1±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±11}{-2}$$

Now solve the equation x=\frac{-1±11}{-2} when ± is plus. Add -1 to 11.

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

Divide 10 by -2.

$$x=-5$$

Now solve the equation x=\frac{-1±11}{-2} when ± is minus. Subtract 11 from -1.

$$x=-\frac{12}{-2}$$

Divide -12 by -2.

$$x=6$$

The equation is now solved.

$$x=-5 x=6$$

Variable x cannot be equal to -5.

$$x=6$$

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.

$$\left(x-5\right)\left(x+6\right)+10=2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x-5 by x+6 and combine like terms.

$$x^{2}+x-30+10=2\left(x-5\right)\left(x+5\right)$$

Add -30 and 10 to get -20.

$$x^{2}+x-20=2\left(x-5\right)\left(x+5\right)$$

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

$$x^{2}+x-20=\left(2x-10\right)\left(x+5\right)$$

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

$$x^{2}+x-20=2x^{2}-50$$

Subtract 2x^{2} from both sides.

$$x^{2}+x-20-2x^{2}=-50$$

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

$$-x^{2}+x-20=-50$$

Add 20 to both sides.

$$-x^{2}+x=-50+20$$

Add -50 and 20 to get -30.

$$-x^{2}+x=-30$$

Divide both sides by -1.

$$\frac{-x^{2}+x}{-1}=-\frac{30}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1}{-1}x=-\frac{30}{-1}$$

Divide 1 by -1.

$$x^{2}-x=-\frac{30}{-1}$$

Divide -30 by -1.

$$x^{2}-x=30$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=30+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=30+\frac{1}{4}$$

Add 30 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{121}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{11}{2} x-\frac{1}{2}=-\frac{11}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=6 x=-5$$

Variable x cannot be equal to -5.

$$x=6$$