Solve -x^2+2x+3/x+3=2

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

Solve for x (complex solution)

$$x=-\sqrt{3}i\approx -0-1.732050808i$$

$$x=\sqrt{3}i\approx 1.732050808i$$

Steps by Finding Square Root

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

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

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

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

Subtract 2x from both sides.

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

Combine 2x and -2x to get 0.

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

Subtract 3 from both sides.

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

Subtract 3 from 6 to get 3.

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

Divide both sides by -1.

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

The equation is now solved.

$$x=\sqrt{3}i x=-\sqrt{3}i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

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

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

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

Subtract 2x from both sides.

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

Combine 2x and -2x to get 0.

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

Subtract 6 from both sides.

$$-x^{2}+3-6=0$$

Subtract 6 from 3 to get -3.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$x=\frac{0±\sqrt{-12}}{2\left(-1\right)}$$

Take the square root of -12.

$$x=\frac{0±2\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±2\sqrt{3}i}{-2}$$

Now solve the equation x=\frac{0±2\sqrt{3}i}{-2} when ± is plus.

$$x=-\sqrt{3}i$$

Now solve the equation x=\frac{0±2\sqrt{3}i}{-2} when ± is minus.

$$x=\sqrt{3}i$$

The equation is now solved.

$$x=-\sqrt{3}i x=\sqrt{3}i$$