Every polynomial equation f(x) = 0 has at least one root, real or imaginary ( complex).

Every polynomial equation f(x) = 0 of degree n has exactly n roots real or imaginary.

A quadratic equation cannot have more than two roots.

If ax^2 + bx + c = 0 is the quadratic equation we get, the roots

\alpha = \frac{-b + \sqrt{b^2 - 4ac} }{2a}   and  \beta = \frac{-b - \sqrt{b^2 - 4ac} }{2a}

If b^2 - 4ac = 0 , then \alpha = \beta  and the roots are \frac{-b}{2a}

If b^2 - 4ac > 0 but is not a perfect square, then the roots are irrational and unequal.

If b^2 - 4ac < 0   then the roots are imaginary and are give by

\alpha = \frac{-b + i\sqrt{4ac - b^2} }{2a}   and  \beta = \frac{-b - i\sqrt{4ac - b^2} }{2a}

Complex roots of an equation with real coefficient  always occur in pairs. However, this may not be true in case of the equations with complex coefficient. For example, x^2 - 2ix - 1 = 0 has both roots equal to i .

Surd root of an equation with rational coefficient  always occurs in pairs like 2 + \sqrt{3} and 2 - \sqrt{3} . However, Surd roots of an equation with irrational coefficients may not occur in pairs. For example, x^2 - 2 \sqrt{3} x + 3 = 0 has both roots equal to \sqrt{3} .