Solve the equation x^2 - 2 sqrt(3) = 0.

Step-by-step solution

1. 1

   Move the constant across. The constant term happens to be irrational, but that changes nothing about the algebra:

   $x^2=2\sqrt{3}$

2. 2

   Check the sign of the right-hand side. $2\sqrt3\approx 3.4641>0$, so real solutions exist. (Had the constant been negative, $x^2$ would have had to be negative and there would be no real root.)

3. 3

   Take square roots with both signs.

   $x=\pm\sqrt{2\sqrt{3}}$

4. 4

   Rewrite the nested radical with fractional exponents. This is the tidiest exact form:

   $\sqrt{2\sqrt3}=\left(2\cdot 3^{1/2}\right)^{1/2}=2^{1/2}\,3^{1/4}=\sqrt{2}\,\sqrt[4]{3}$

   Numerically $\sqrt2\approx 1.41421$ and $\sqrt[4]3\approx 1.31607$, giving $x\approx\pm 1.8612$.

5. 5

   Verify by squaring back. $1.8612^2\approx 3.4641$ and $2\sqrt3\approx 3.4641$ $\checkmark$. The two roots are exact opposites, as always for an equation of the form $x^2=k$.