We know the formula to calculate roots of the quadratic equation \(a{x^2} + bx + c = 0\)\[\color{blue}{x=\frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}}\]

Nature of the roots is determined by the value of \({b^2} - 4ac\),  calculating this value we can determine whether the roots of the quadratic equation \(a{x^2} + bx + c = 0\) are real/complex, rational/irrational, equal/Distinct.

• If \({b^2} - 4ac > 0\), then the roots are real and distinct.
• If \({b^2} - 4ac < 0\), then the roots are complex numbers.
• If \({b^2} - 4ac = 0\), then the roots are real and equal.
• If \({b^2} - 4ac\) is a perfect square and \(a,\;b,\;c\) are rational numbers, then the roots are real and rational numbers. 

Sum and product of the roots:

Further if the roots of the equation \(a{x^2} + bx + c = 0\), are \(\alpha ,\;\beta \), then sum and product of the roots is given by:\[\alpha  + \beta  =  - \frac{b}{a}\] \[\alpha \beta  = \frac{c}{a}\]

Example: If the roots of the equation \(ax^2+bx+c=0\) are \(\alpha\) and \(\beta\), then find the value of \({\alpha ^3} + {\beta ^3}\).

Solution: We know that sum of the roots, \(\alpha  + \beta  = p\)and product of the roots \(\alpha  \cdot \beta  = q\)

Now factorize \({\alpha ^3} + {\beta ^3} = (\alpha  + \beta )({\alpha ^2} + {\beta ^2} - \alpha \beta )\)

\( = (\alpha  + \beta )[{(\alpha  + \beta )^2} - 3\alpha \beta ]\)

\( = p({p^2} - 3q)\)