Solve y=sqrt{6+y}

$$y = \sqrt { 6 + y }$$

Solve for y

$$y=3$$

Solution Steps

Square both sides of the equation.

$$y^{2}=\left(\sqrt{6+y}\right)^{2}$$

Calculate \sqrt{6+y} to the power of 2 and get 6+y.

$$y^{2}=6+y$$

Subtract 6 from both sides.

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

Subtract y from both sides.

$$y^{2}-6-y=0$$

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

$$y^{2}-y-6=0$$

To solve the equation, factor y^{2}-y-6 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.

$$a+b=-1 ab=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -1.

$$a=-3 b=2$$

Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.

$$\left(y-3\right)\left(y+2\right)$$

To find equation solutions, solve y-3=0 and y+2=0.

$$y=3 y=-2$$

Substitute 3 for y in the equation y=\sqrt{6+y}.

$$3=\sqrt{6+3}$$

Simplify. The value y=3 satisfies the equation.

$$3=3$$

Substitute -2 for y in the equation y=\sqrt{6+y}.

$$-2=\sqrt{6-2}$$

Simplify. The value y=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Equation y=\sqrt{y+6} has a unique solution.

$$y=3$$