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

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

Solve for x (complex solution)

$$x=-i\sqrt{\sqrt{3}-1}\approx -0-0.855599677i$$

$$x=i\sqrt{\sqrt{3}-1}\approx 0.855599677i$$

Steps by Finding Square Root

Add 1 to both sides.

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

The equation is now solved.

$$x=i\sqrt{\sqrt{3}-1} x=-i\sqrt{\sqrt{3}-1}$$

Steps Using the Quadratic Formula

Add \sqrt{3} to both sides.

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

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.

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

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

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

Square 0.

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

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

$$x=\frac{0±\sqrt{4-4\sqrt{3}}}{2}$$

Take the square root of 4-4\sqrt{3}.

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

Now solve the equation x=\frac{0±2i\sqrt{-\left(1-\sqrt{3}\right)}}{2} when ± is plus.

$$x=i\sqrt{\sqrt{3}-1}$$

Now solve the equation x=\frac{0±2i\sqrt{-\left(1-\sqrt{3}\right)}}{2} when ± is minus.

$$x=-i\sqrt{\sqrt{3}-1}$$

The equation is now solved.

$$x=i\sqrt{\sqrt{3}-1} x=-i\sqrt{\sqrt{3}-1}$$