Solve x/x+3+6/x^2+8x+15=2

$$\frac { x } { x + 3 } + \frac { 6 } { x ^ { 2 } + 8 x + 15 } = 2$$

Solve for x

$$x=-8$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)\left(x+5\right), the least common multiple of x+3,x^{2}+8x+15.

$$\left(x+5\right)x+6=2\left(x+3\right)\left(x+5\right)$$

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

$$x^{2}+5x+6=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x+3.

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

Use the distributive property to multiply 2x+6 by x+5 and combine like terms.

$$x^{2}+5x+6=2x^{2}+16x+30$$

Subtract 2x^{2} from both sides.

$$x^{2}+5x+6-2x^{2}=16x+30$$

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

$$-x^{2}+5x+6=16x+30$$

Subtract 16x from both sides.

$$-x^{2}+5x+6-16x=30$$

Combine 5x and -16x to get -11x.

$$-x^{2}-11x+6=30$$

Subtract 30 from both sides.

$$-x^{2}-11x+6-30=0$$

Subtract 30 from 6 to get -24.

$$-x^{2}-11x-24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-11 ab=-\left(-24\right)=24$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-3 b=-8$$

Rewrite -x^{2}-11x-24 as \left(-x^{2}-3x\right)+\left(-8x-24\right).

$$\left(-x^{2}-3x\right)+\left(-8x-24\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(-x-3\right)+8\left(-x-3\right)$$

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

$$\left(-x-3\right)\left(x+8\right)$$

To find equation solutions, solve -x-3=0 and x+8=0.

$$x=-3 x=-8$$

Variable x cannot be equal to -3.

$$x=-8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)\left(x+5\right), the least common multiple of x+3,x^{2}+8x+15.

$$\left(x+5\right)x+6=2\left(x+3\right)\left(x+5\right)$$

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

$$x^{2}+5x+6=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x+3.

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

Use the distributive property to multiply 2x+6 by x+5 and combine like terms.

$$x^{2}+5x+6=2x^{2}+16x+30$$

Subtract 2x^{2} from both sides.

$$x^{2}+5x+6-2x^{2}=16x+30$$

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

$$-x^{2}+5x+6=16x+30$$

Subtract 16x from both sides.

$$-x^{2}+5x+6-16x=30$$

Combine 5x and -16x to get -11x.

$$-x^{2}-11x+6=30$$

Subtract 30 from both sides.

$$-x^{2}-11x+6-30=0$$

Subtract 30 from 6 to get -24.

$$-x^{2}-11x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -11 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-11\right)±\sqrt{121+4\left(-24\right)}}{2\left(-1\right)}$$

Multiply 4 times -24.

$$x=\frac{-\left(-11\right)±\sqrt{121-96}}{2\left(-1\right)}$$

Add 121 to -96.

$$x=\frac{-\left(-11\right)±\sqrt{25}}{2\left(-1\right)}$$

Take the square root of 25.

$$x=\frac{-\left(-11\right)±5}{2\left(-1\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{11±5}{-2}$$

Now solve the equation x=\frac{11±5}{-2} when ± is plus. Add 11 to 5.

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{11±5}{-2} when ± is minus. Subtract 5 from 11.

$$x=\frac{6}{-2}$$

Divide 6 by -2.

$$x=-3$$

The equation is now solved.

$$x=-8 x=-3$$

Variable x cannot be equal to -3.

$$x=-8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)\left(x+5\right), the least common multiple of x+3,x^{2}+8x+15.

$$\left(x+5\right)x+6=2\left(x+3\right)\left(x+5\right)$$

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

$$x^{2}+5x+6=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x+3.

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

Use the distributive property to multiply 2x+6 by x+5 and combine like terms.

$$x^{2}+5x+6=2x^{2}+16x+30$$

Subtract 2x^{2} from both sides.

$$x^{2}+5x+6-2x^{2}=16x+30$$

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

$$-x^{2}+5x+6=16x+30$$

Subtract 16x from both sides.

$$-x^{2}+5x+6-16x=30$$

Combine 5x and -16x to get -11x.

$$-x^{2}-11x+6=30$$

Subtract 6 from both sides.

$$-x^{2}-11x=30-6$$

Subtract 6 from 30 to get 24.

$$-x^{2}-11x=24$$

Divide both sides by -1.

$$\frac{-x^{2}-11x}{-1}=\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{11}{-1}\right)x=\frac{24}{-1}$$

Divide -11 by -1.

$$x^{2}+11x=\frac{24}{-1}$$

Divide 24 by -1.

$$x^{2}+11x=-24$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-24+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=-24+\frac{121}{4}$$

Add -24 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{25}{4}$$

Factor x^{2}+11x+\frac{121}{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}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{5}{2} x+\frac{11}{2}=-\frac{5}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=-3 x=-8$$

Variable x cannot be equal to -3.

$$x=-8$$