let ax2 + bx + c =0,where a,b,c are all positive real numbers ,then what is the nature of their roots?

Answer :

We have quadratic equation ax² +  bx  +  c  = 0  , Where a , b and c are positive  numbers .

SO , we find roots of this equation by completing the square method , As :

$\Rightarrow$ax² +  bx  = - c 

$$\begin{gathered} \Rightarrow x^2 + \frac{b}{a}x = -\frac{c}{a} \\ Now we add both side term by taking half of x - term , and square it \\ \Rightarrow x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}= -\frac{c}{a}+ \frac{b^2}{4a^2} \\ Taking L.C.M. on R.H.S. we get \\ \Rightarrow x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}= \frac{- 4ac +\hspace{0.17em}{b}^2}{4a^2} \\ \Rightarrow {\left(x + \frac{b}{2a}\right)}^2 = \frac{\hspace{0.17em}{b}^2- 4ac}{4a^2} \\ \Rightarrow x + \frac{b}{2a}= \pm \sqrt{\frac{\hspace{0.17em}{b}^2- 4ac}{4a^2}} \\ \Rightarrow x + \frac{b}{2a}= \pm \frac{\sqrt{\hspace{0.17em}{b}^2- 4ac}}{2a} \\ \Rightarrow x = - \frac{b}{2a}\pm \frac{\sqrt{\hspace{0.17em}{b}^2- 4ac}}{2a} \\ \Rightarrow x = \frac{- b \pm \sqrt{\hspace{0.17em}{b}^2- 4ac}}{2a} , SO \\ Roots , are \\ \mathbf{\Rightarrow }\left(\mathbf{x}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{-}\mathbf{ }\mathbf{b}\mathbf{ }\mathbf{+}\sqrt{\mathbf{\hspace{0.17em}}{\mathbf{b}}^{\mathbf{2}}\mathbf{-}\mathbf{ }\mathbf{4}\mathbf{a}\mathbf{c}}}{\mathbf{2}\mathbf{a}}\mathbf{ }\right)\mathbf{ }\mathit{A}\mathit{n}\mathit{d}\mathbf{ }\left(\mathbf{x}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{-}\mathbf{ }\mathbf{b}\mathbf{ }\mathbf{-}\sqrt{\mathbf{\hspace{0.17em}}{\mathbf{b}}^{\mathbf{2}}\mathbf{-}\mathbf{ }\mathbf{4}\mathbf{a}\mathbf{c}}}{\mathbf{2}\mathbf{a}}\mathbf{ }\right)\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ } \end{gathered}$$

Here , Values of a . b and c could be any positive numbers , So

Nature of Roots depends on  value of $\sqrt{b^2 - 4ac}$ , As :

1 ) If $\sqrt{b^2 - 4ac}$ <  0  Then roots are imaginary .
2 ) If $\sqrt{b^2 - 4ac}$ > 0 , There are two real roots , and we have two possibilities , As :
 i ) If $\sqrt{b^2 - 4ac}$  is a perfect square than we have rational roots .
ii ) If $\sqrt{b^2 - 4ac}$ is not a perfect square than we have irrational roots .
3 ) If $\sqrt{b^2 - 4ac}$  = 0 , Roots are equal or in other words we have only one root