Solve y^2+2y+3/y+1=2

$$\frac{ { y }^{ 2 } +2y+3 }{ y+1 } =2$$

Solve for y

$$y=-i$$

$$y=i$$

Steps by Finding Square Root

Variable y cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by y+1.

$$y^{2}+2y+3=2\left(y+1\right)$$

Use the distributive property to multiply 2 by y+1.

$$y^{2}+2y+3=2y+2$$

Subtract 2y from both sides.

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

Combine 2y and -2y to get 0.

$$y^{2}+3=2$$

Subtract 3 from both sides.

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

Subtract 3 from 2 to get -1.

$$y^{2}=-1$$

The equation is now solved.

$$y=i y=-i$$

Steps Using the Quadratic Formula

Variable y cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by y+1.

$$y^{2}+2y+3=2\left(y+1\right)$$

Use the distributive property to multiply 2 by y+1.

$$y^{2}+2y+3=2y+2$$

Subtract 2y from both sides.

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

Combine 2y and -2y to get 0.

$$y^{2}+3=2$$

Subtract 2 from both sides.

$$y^{2}+3-2=0$$

Subtract 2 from 3 to get 1.

$$y^{2}+1=0$$

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

$$y=\frac{0±\sqrt{0^{2}-4}}{2}$$

Square 0.

$$y=\frac{0±\sqrt{-4}}{2}$$

Take the square root of -4.

$$y=\frac{0±2i}{2}$$

Now solve the equation y=\frac{0±2i}{2} when ± is plus.

$$y=i$$

Now solve the equation y=\frac{0±2i}{2} when ± is minus.

$$y=-i$$

The equation is now solved.

$$y=i y=-i$$