QUESTION

# If l, m are real $l\ne m$ then the roots of the equation $\left( l-m \right){{x}^{2}}-5\left( l+m \right)x-2\left( l-m \right)=0$ are:(a) Real and equal(b) Non-real and complex(c) Real and unequal(d) None of these

Hint: For solving this problem we will use the direct formulae of the discriminant of the quadratic equation and we can find the nature of the roots of the quadratic equation.

Given:
We have a quadratic equation $\left( l-m \right){{x}^{2}}-5\left( l+m \right)x-2\left( l-m \right)=0$ where $l$ and $m$ are real and $l\ne m$ .
Now, before we proceed we should see the formula of the discriminant of the quadratic equation and its impact on the roots of the quadratic equation. Formulae are given below:
Quadratic equation: $a{{x}^{2}}+bx+c=0$ . Then, the roots of the equation are: $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
The discriminant of the quadratic equation $=D={{b}^{2}}-4ac$ .
Now, if $D>0$ then, roots will be real and distinct. If $D=0$ then, roots will be real and equal. If $D<0$ then, roots will be unequal and not real.
Now, in the question we have: $\left( l-m \right){{x}^{2}}-5\left( l+m \right)x-2\left( l-m \right)=0$

So, $a=\left( l-m \right)$ , $b=-5\left( l+m \right)$ and $c=-2\left( l-m \right)$

Then, \begin{align} & D={{b}^{2}}-4ac={{\left( -5\left( l+m \right) \right)}^{2}}-4\left( l-m \right)\times \left( -2\left( l-m \right) \right) \\ & \Rightarrow D=25{{\left( l+m \right)}^{2}}+8{{\left( l-m \right)}^{2}} \\ \end{align}

Thus, the discriminant of the given equation will be $D=25{{\left( l+m \right)}^{2}}+8{{\left( l-m \right)}^{2}}$
Now, in the above expression of discriminant, we can say that both terms ${{\left( l+m \right)}^{2}}$ and ${{\left( l-m \right)}^{2}}$ are greater than zero. Thus, the discriminant will be greater than zero $\left( D>0 \right)$ .
Now, as we know that when the discriminant of the quadratic equation is greater than 0 then roots will be real and unequal.
Thus, the roots of $\left( l-m \right){{x}^{2}}-5\left( l+m \right)x-2\left( l-m \right)=0$ will be real and unequal.
Hence, (c) is the correct option.

Note: The question was very easy to solve only we should know how we can tell about the nature of roots from the value of discriminant of the quadratic equation. But the student should avoid calculation mistakes in calculation of discriminant and apply the conditions correctly to get the correct answer.