Solve y=sqrt{5y+6}

$$y= \sqrt{ 5y+6 }$$

Solve for y

$$y=6$$

Solution Steps

Square both sides of the equation.

$$y^{2}=\left(\sqrt{5y+6}\right)^{2}$$

Calculate \sqrt{5y+6} to the power of 2 and get 5y+6.

$$y^{2}=5y+6$$

Subtract 5y from both sides.

$$y^{2}-5y=6$$

Subtract 6 from both sides.

$$y^{2}-5y-6=0$$

To solve the equation, factor y^{2}-5y-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.

$$a+b=-5 ab=-6$$

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 -5.

$$a=-6 b=1$$

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

$$\left(y-6\right)\left(y+1\right)$$

To find equation solutions, solve y-6=0 and y+1=0.

$$y=6 y=-1$$

Substitute 6 for y in the equation y=\sqrt{5y+6}.

$$6=\sqrt{5\times 6+6}$$

Simplify. The value y=6 satisfies the equation.

$$6=6$$

Substitute -1 for y in the equation y=\sqrt{5y+6}.

$$-1=\sqrt{5\left(-1\right)+6}$$

Simplify. The value y=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation y=\sqrt{5y+6} has a unique solution.

$$y=6$$