Solve x^2-4sqrt{3}+12=0

$${ x }^{ 2 } -4 \sqrt{ 3 } +12=0$$

Solve for x (complex solution)

$$x=-2i\sqrt{3-\sqrt{3}}\approx -0-2.252065001i$$

$$x=2i\sqrt{3-\sqrt{3}}\approx 2.252065001i$$

Steps by Finding Square Root

Add 4\sqrt{3} to both sides. Anything plus zero gives itself.

$$x^{2}+12=4\sqrt{3}$$

Subtract 12 from both sides.

$$x^{2}=4\sqrt{3}-12$$

The equation is now solved.

$$x=2i\sqrt[4]{3}\sqrt{\sqrt{3}-1} x=-2i\sqrt[4]{3}\sqrt{\sqrt{3}-1}$$

Steps Using the Quadratic Formula

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.

$$x^{2}+12-4\sqrt{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -4\sqrt{3}+12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(12-4\sqrt{3}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(12-4\sqrt{3}\right)}}{2}$$

Multiply -4 times -4\sqrt{3}+12.

$$x=\frac{0±\sqrt{16\sqrt{3}-48}}{2}$$

Take the square root of 16\sqrt{3}-48.

$$x=\frac{0±4i\sqrt{3-\sqrt{3}}}{2}$$

Now solve the equation x=\frac{0±4i\sqrt{3-\sqrt{3}}}{2} when ± is plus.

$$x=2i\sqrt[4]{3}\sqrt{\sqrt{3}-1}$$

Now solve the equation x=\frac{0±4i\sqrt{3-\sqrt{3}}}{2} when ± is minus.

$$x=-2i\sqrt[4]{3}\sqrt{\sqrt{3}-1}$$

The equation is now solved.

$$x=2i\sqrt[4]{3}\sqrt{\sqrt{3}-1} x=-2i\sqrt[4]{3}\sqrt{\sqrt{3}-1}$$