Solve x+6=sqrt{x+6}

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

Solve for x

$$x=-6$$

$$x=-5$$

Solution Steps

Square both sides of the equation.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36=\left(\sqrt{x+6}\right)^{2}$$

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

$$x^{2}+12x+36=x+6$$

Subtract x from both sides.

$$x^{2}+12x+36-x=6$$

Combine 12x and -x to get 11x.

$$x^{2}+11x+36=6$$

Subtract 6 from both sides.

$$x^{2}+11x+36-6=0$$

Subtract 6 from 36 to get 30.

$$x^{2}+11x+30=0$$

To solve the equation, factor x^{2}+11x+30 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.

$$a+b=11 ab=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 11.

$$a=5 b=6$$

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

$$\left(x+5\right)\left(x+6\right)$$

To find equation solutions, solve x+5=0 and x+6=0.

$$x=-5 x=-6$$

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

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

Simplify. The value x=-5 satisfies the equation.

$$1=1$$

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

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

Simplify. The value x=-6 satisfies the equation.

$$0=0$$

List all solutions of x+6=\sqrt{x+6}.

$$x=-5 x=-6$$