Solve x^2=1/2-2sqrt{3}

$${ x }^{ 2 } = \frac{ 1 }{ 2 } -2 \sqrt{ 3 }$$

Solve for x (complex solution)

$$x=-\frac{i\sqrt{8\sqrt{3}-2}}{2}\approx -0-1.721656648i$$

$$x=\frac{i\sqrt{8\sqrt{3}-2}}{2}\approx 1.721656648i$$

Steps by Finding Square Root

The equation is now solved.

$$x=\frac{i\sqrt{8\sqrt{3}-2}}{2} x=-\frac{i\sqrt{8\sqrt{3}-2}}{2}$$

Steps Using the Quadratic Formula

Subtract \frac{1}{2} from both sides.

$$x^{2}-\frac{1}{2}=-2\sqrt{3}$$

Add 2\sqrt{3} to both sides.

$$x^{2}-\frac{1}{2}+2\sqrt{3}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$x^{2}+2\sqrt{3}-\frac{1}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{1}{2}+2\sqrt{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(2\sqrt{3}-\frac{1}{2}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(2\sqrt{3}-\frac{1}{2}\right)}}{2}$$

Multiply -4 times -\frac{1}{2}+2\sqrt{3}.

$$x=\frac{0±\sqrt{2-8\sqrt{3}}}{2}$$

Take the square root of 2-8\sqrt{3}.

$$x=\frac{0±i\sqrt{8\sqrt{3}-2}}{2}$$

Now solve the equation x=\frac{0±i\sqrt{8\sqrt{3}-2}}{2} when ± is plus.

$$x=\frac{i\sqrt{8\sqrt{3}-2}}{2}$$

Now solve the equation x=\frac{0±i\sqrt{8\sqrt{3}-2}}{2} when ± is minus.

$$x=-\frac{i\sqrt{8\sqrt{3}-2}}{2}$$

The equation is now solved.

$$x=\frac{i\sqrt{8\sqrt{3}-2}}{2} x=-\frac{i\sqrt{8\sqrt{3}-2}}{2}$$