Solve x^2-5-2sqrt{6}=0

$$x ^ { 2 } - 5 - 2 \sqrt { 6 } = 0$$

Solve for x (complex solution)

$$x=-\left(\sqrt{2}+\sqrt{3}\right)\approx -3.14626437$$

$$x=\sqrt{2}+\sqrt{3}\approx 3.14626437$$

Steps by Finding Square Root

Add 5 to both sides. Anything plus zero gives itself.

$$x^{2}-2\sqrt{6}=5$$

Add 2\sqrt{6} to both sides.

$$x^{2}=5+2\sqrt{6}$$

The equation is now solved.

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

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.

$$x^{2}-2\sqrt{6}-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -5-2\sqrt{6} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-2\sqrt{6}-5\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2\sqrt{6}-5\right)}}{2}$$

Multiply -4 times -5-2\sqrt{6}.

$$x=\frac{0±\sqrt{8\sqrt{6}+20}}{2}$$

Take the square root of 20+8\sqrt{6}.

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

Now solve the equation x=\frac{0±\left(2\sqrt{2}+2\sqrt{3}\right)}{2} when ± is plus.

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

Now solve the equation x=\frac{0±\left(2\sqrt{2}+2\sqrt{3}\right)}{2} when ± is minus.

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

The equation is now solved.

$$x=\sqrt{2}+\sqrt{3} x=-\sqrt{2}-\sqrt{3}$$

Solve for x

$$x=\sqrt{2}+\sqrt{3}\approx 3.14626437$$

$$x=-\sqrt{2}-\sqrt{3}\approx -3.14626437$$