Solve y-sqrt{y+2}=0

$$y - \sqrt { y + 2 } = 0$$

Solve for y

$$y=2$$

Solution Steps

Subtract y from both sides of the equation.

$$-\sqrt{y+2}=-y$$

Cancel out -1 on both sides.

$$\sqrt{y+2}=y$$

Square both sides of the equation.

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

Calculate \sqrt{y+2} to the power of 2 and get y+2.

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

Subtract y^{2} from both sides.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

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

$$a+b=1 ab=-2=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=2 b=-1$$

Rewrite -y^{2}+y+2 as \left(-y^{2}+2y\right)+\left(-y+2\right).

$$\left(-y^{2}+2y\right)+\left(-y+2\right)$$

Factor out -y in the first and -1 in the second group.

$$-y\left(y-2\right)-\left(y-2\right)$$

Factor out common term y-2 by using distributive property.

$$\left(y-2\right)\left(-y-1\right)$$

To find equation solutions, solve y-2=0 and -y-1=0.

$$y=2 y=-1$$

Substitute 2 for y in the equation y-\sqrt{y+2}=0.

$$2-\sqrt{2+2}=0$$

Simplify. The value y=2 satisfies the equation.

$$0=0$$

Substitute -1 for y in the equation y-\sqrt{y+2}=0.

$$-1-\sqrt{-1+2}=0$$

Simplify. The value y=-1 does not satisfy the equation.

$$-2=0$$

Equation \sqrt{y+2}=y has a unique solution.

$$y=2$$