Solve (x)=sqrt{2+x}

$$( x ) = \sqrt { 2 + x }$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

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

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

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

Subtract 2 from both sides.

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

Subtract x from both sides.

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

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

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

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

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

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. The only such pair is the system solution.

$$a=-2 b=1$$

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

$$\left(x-2\right)\left(x+1\right)$$

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

$$x=2 x=-1$$

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

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

Simplify. The value x=2 satisfies the equation.

$$2=2$$

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

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

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

$$-1=1$$

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

$$x=2$$