Solve x+2=sqrt{4x+13}

$$x + 2 = \sqrt { 4 x + 13 }$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(x+2\right)^{2}=\left(\sqrt{4x+13}\right)^{2}$$

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

$$x^{2}+4x+4=\left(\sqrt{4x+13}\right)^{2}$$

Calculate \sqrt{4x+13} to the power of 2 and get 4x+13.

$$x^{2}+4x+4=4x+13$$

Subtract 4x from both sides.

$$x^{2}+4x+4-4x=13$$

Combine 4x and -4x to get 0.

$$x^{2}+4=13$$

Subtract 13 from both sides.

$$x^{2}+4-13=0$$

Subtract 13 from 4 to get -9.

$$x^{2}-9=0$$

Consider x^{2}-9. Rewrite x^{2}-9 as x^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-3\right)\left(x+3\right)=0$$

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

$$x=3 x=-3$$

Substitute 3 for x in the equation x+2=\sqrt{4x+13}.

$$3+2=\sqrt{4\times 3+13}$$

Simplify. The value x=3 satisfies the equation.

$$5=5$$

Substitute -3 for x in the equation x+2=\sqrt{4x+13}.

$$-3+2=\sqrt{4\left(-3\right)+13}$$

Simplify. The value x=-3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation x+2=\sqrt{4x+13} has a unique solution.

$$x=3$$