Solve 2x+1/x^2+6x+9-x-2/x^2+3x=x^2+6/x^3-9x

$$\frac { 2 x + 1 } { x ^ { 2 } + 6 x + 9 } - \frac { x - 2 } { x ^ { 2 } + 3 x } = \frac { x ^ { 2 } + 6 } { x ^ { 3 } - 9 x }$$

Solve for x (complex solution)

$$x=-\sqrt{6}i\approx -0-2.449489743i$$

$$x=\sqrt{6}i\approx 2.449489743i$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right)\left(x+3\right)^{2}, the least common multiple of x^{2}+6x+9,x^{2}+3x,x^{3}-9x.

$$x\left(x-3\right)\left(2x+1\right)-\left(x^{2}-9\right)\left(x-2\right)=\left(x+3\right)\left(x^{2}+6\right)$$

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

$$\left(x^{2}-3x\right)\left(2x+1\right)-\left(x^{2}-9\right)\left(x-2\right)=\left(x+3\right)\left(x^{2}+6\right)$$

Use the distributive property to multiply x^{2}-3x by 2x+1 and combine like terms.

$$2x^{3}-5x^{2}-3x-\left(x^{2}-9\right)\left(x-2\right)=\left(x+3\right)\left(x^{2}+6\right)$$

Use the distributive property to multiply x^{2}-9 by x-2.

$$2x^{3}-5x^{2}-3x-\left(x^{3}-2x^{2}-9x+18\right)=\left(x+3\right)\left(x^{2}+6\right)$$

To find the opposite of x^{3}-2x^{2}-9x+18, find the opposite of each term.

$$2x^{3}-5x^{2}-3x-x^{3}+2x^{2}+9x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Combine 2x^{3} and -x^{3} to get x^{3}.

$$x^{3}-5x^{2}-3x+2x^{2}+9x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Combine -5x^{2} and 2x^{2} to get -3x^{2}.

$$x^{3}-3x^{2}-3x+9x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Combine -3x and 9x to get 6x.

$$x^{3}-3x^{2}+6x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Use the distributive property to multiply x+3 by x^{2}+6.

$$x^{3}-3x^{2}+6x-18=x^{3}+6x+3x^{2}+18$$

Subtract x^{3} from both sides.

$$x^{3}-3x^{2}+6x-18-x^{3}=6x+3x^{2}+18$$

Combine x^{3} and -x^{3} to get 0.

$$-3x^{2}+6x-18=6x+3x^{2}+18$$

Subtract 6x from both sides.

$$-3x^{2}+6x-18-6x=3x^{2}+18$$

Combine 6x and -6x to get 0.

$$-3x^{2}-18=3x^{2}+18$$

Subtract 3x^{2} from both sides.

$$-3x^{2}-18-3x^{2}=18$$

Combine -3x^{2} and -3x^{2} to get -6x^{2}.

$$-6x^{2}-18=18$$

Add 18 to both sides.

$$-6x^{2}=18+18$$

Add 18 and 18 to get 36.

$$-6x^{2}=36$$

Divide both sides by -6.

$$x^{2}=\frac{36}{-6}$$

Divide 36 by -6 to get -6.

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

The equation is now solved.

$$x=\sqrt{6}i x=-\sqrt{6}i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right)\left(x+3\right)^{2}, the least common multiple of x^{2}+6x+9,x^{2}+3x,x^{3}-9x.

$$x\left(x-3\right)\left(2x+1\right)-\left(x^{2}-9\right)\left(x-2\right)=\left(x+3\right)\left(x^{2}+6\right)$$

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

$$\left(x^{2}-3x\right)\left(2x+1\right)-\left(x^{2}-9\right)\left(x-2\right)=\left(x+3\right)\left(x^{2}+6\right)$$

Use the distributive property to multiply x^{2}-3x by 2x+1 and combine like terms.

$$2x^{3}-5x^{2}-3x-\left(x^{2}-9\right)\left(x-2\right)=\left(x+3\right)\left(x^{2}+6\right)$$

Use the distributive property to multiply x^{2}-9 by x-2.

$$2x^{3}-5x^{2}-3x-\left(x^{3}-2x^{2}-9x+18\right)=\left(x+3\right)\left(x^{2}+6\right)$$

To find the opposite of x^{3}-2x^{2}-9x+18, find the opposite of each term.

$$2x^{3}-5x^{2}-3x-x^{3}+2x^{2}+9x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Combine 2x^{3} and -x^{3} to get x^{3}.

$$x^{3}-5x^{2}-3x+2x^{2}+9x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Combine -5x^{2} and 2x^{2} to get -3x^{2}.

$$x^{3}-3x^{2}-3x+9x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Combine -3x and 9x to get 6x.

$$x^{3}-3x^{2}+6x-18=\left(x+3\right)\left(x^{2}+6\right)$$

Use the distributive property to multiply x+3 by x^{2}+6.

$$x^{3}-3x^{2}+6x-18=x^{3}+6x+3x^{2}+18$$

Subtract x^{3} from both sides.

$$x^{3}-3x^{2}+6x-18-x^{3}=6x+3x^{2}+18$$

Combine x^{3} and -x^{3} to get 0.

$$-3x^{2}+6x-18=6x+3x^{2}+18$$

Subtract 6x from both sides.

$$-3x^{2}+6x-18-6x=3x^{2}+18$$

Combine 6x and -6x to get 0.

$$-3x^{2}-18=3x^{2}+18$$

Subtract 3x^{2} from both sides.

$$-3x^{2}-18-3x^{2}=18$$

Combine -3x^{2} and -3x^{2} to get -6x^{2}.

$$-6x^{2}-18=18$$

Subtract 18 from both sides.

$$-6x^{2}-18-18=0$$

Subtract 18 from -18 to get -36.

$$-6x^{2}-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 0 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-6\right)\left(-36\right)}}{2\left(-6\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-6\right)\left(-36\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{0±\sqrt{24\left(-36\right)}}{2\left(-6\right)}$$

Multiply 24 times -36.

$$x=\frac{0±\sqrt{-864}}{2\left(-6\right)}$$

Take the square root of -864.

$$x=\frac{0±12\sqrt{6}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{0±12\sqrt{6}i}{-12}$$

Now solve the equation x=\frac{0±12\sqrt{6}i}{-12} when ± is plus.

$$x=-\sqrt{6}i$$

Now solve the equation x=\frac{0±12\sqrt{6}i}{-12} when ± is minus.

$$x=\sqrt{6}i$$

The equation is now solved.

$$x=-\sqrt{6}i x=\sqrt{6}i$$