Mathematics - Quadratic Equation Question with Solution | TestHub

$f(x)=\left(x-a_1\right)\left(x-a_2\right)\left(x-a_2\right)+\left(a_1-x\right)+\left(a_2-x\right)+\left(a_3-x\right)$

Now, $f(x)\to -\infty$ as $x\to -\infty$ and $f(x)\to \infty$ are $x\to \infty$

Again $f\left(a_1\right)=\left(a_2-a_1\right)+\left(a_3-a_1\right)>0\left[\because a_1<a_2<a_3\right]$

⇒ One root belongs to $\left(-\infty ,a_1\right)$

Also, $f\left(a_3\right)=\left(a_1-a_3\right)+\left(a_2-a_3\right)<0$

⇒ One root belongs to $\left(a_1,a_3\right)$

So, $f(x)=0$ has three distinct real roots.