Solve x=sqrt{(x+6)}

$$x= \sqrt{ (x+6) }$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

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

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

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

Subtract x from both sides.

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

Subtract 6 from both sides.

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

To solve the equation, factor x^{2}-x-6 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+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(x+a\right)\left(x+b\right) using the obtained values.

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

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

$$x=3 x=-2$$

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

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

Simplify. The value x=3 satisfies the equation.

$$3=3$$

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

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

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

$$-2=2$$

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

$$x=3$$