Solve 5/x^2+5x+6+x/x+2=1/x+3

$$\frac { 5 } { x ^ { 2 } + 5 x + 6 } + \frac { x } { x + 2 } = \frac { 1 } { x + 3 }$$

Solve for x (complex solution)

$$x=-1+\sqrt{2}i\approx -1+1.414213562i$$

$$x=-\sqrt{2}i-1\approx -1-1.414213562i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of x^{2}+5x+6,x+2,x+3.

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

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

$$5+x^{2}+3x=x+2$$

Subtract x from both sides.

$$5+x^{2}+3x-x=2$$

Combine 3x and -x to get 2x.

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

Subtract 2 from both sides.

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

Subtract 2 from 5 to get 3.

$$3+x^{2}+2x=0$$

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.

$$x^{2}+2x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-2±\sqrt{4-12}}{2}$$

Add 4 to -12.

$$x=\frac{-2±\sqrt{-8}}{2}$$

Take the square root of -8.

$$x=\frac{-2±2\sqrt{2}i}{2}$$

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

$$x=\frac{-2+2\sqrt{2}i}{2}$$

Divide -2+2i\sqrt{2} by 2.

$$x=-1+\sqrt{2}i$$

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

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

Divide -2-2i\sqrt{2} by 2.

$$x=-\sqrt{2}i-1$$

The equation is now solved.

$$x=-1+\sqrt{2}i x=-\sqrt{2}i-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of x^{2}+5x+6,x+2,x+3.

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

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

$$5+x^{2}+3x=x+2$$

Subtract x from both sides.

$$5+x^{2}+3x-x=2$$

Combine 3x and -x to get 2x.

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

Subtract 5 from both sides.

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

Subtract 5 from 2 to get -3.

$$x^{2}+2x=-3$$

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

$$x^{2}+2x+1^{2}=-3+1^{2}$$

Square 1.

$$x^{2}+2x+1=-3+1$$

Add -3 to 1.

$$x^{2}+2x+1=-2$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{2}i x+1=-\sqrt{2}i$$

Subtract 1 from both sides of the equation.

$$x=-1+\sqrt{2}i x=-\sqrt{2}i-1$$