Let \(ax^{2}+bx+c=0\) is a quadratic equation
\(\alpha\) and \(\beta\) are two roots of this equation
Then, sum of roots =\(\alpha+\beta=\frac{-b}{a}\)
And, product of roots =\(\alpha\beta=\frac{c}{a}\)
But we have to find difference between roots,
\(\alpha-\beta=\sqrt{(\alpha-\beta)^{2}}\)
\(\alpha-\beta=\sqrt{\alpha^{2}+\beta^{2}-2\alpha\beta}\)
\(\alpha-\beta=\sqrt{(\alpha+\beta)^{2}-2\alpha\beta}\)
Substitute the values of \(\alpha+\beta\) and \(\alpha\beta\)
\(\alpha-\beta=\sqrt{(\frac{-b}{a})^{2}-4(\frac{c}{a})}\)
\(\alpha-\beta=\frac{\sqrt{b^{2}-4ac}}{a}\)
We know that , \(D=\sqrt{b^{2}-4ac}\)
\(\alpha-\beta=\frac{\sqrt{D}}{a}\)
