Question

If $l,m,n$ are real, $l\ne m$, then the roots by the equation: $(l-m)x^2-5(l+m)x-2(l-m)=0$ are:

• A. Complex
• B. Real and Equal
• C. Real and unequal.
• D. None of these

Answer 1

The correct option is C

Real and unequal.

Explanation for the correct option.

By comparing the given quadratic equation, with the general equation we get,

$a=\left(l-m\right),b=-5(l+m),\mathrm{and}c=-2\left(l-m\right)$.

So, the discriminant will be,

$\begin{array}{rcl}D& =& b^2-4ac\\ & =& 25{\left(l+m\right)}^2-4\left(l-m\right)\left(-2\right)\left(l-m\right)\\ & =& 25{\left(l+m\right)}^2+8{\left(l-m\right)}^2\end{array}$

As, $l\mathrm{and}m$ are real, $l\ne m$, so ${\left(l+m\right)}^{2}and{\left(l-m\right)}^2>0$

$\Rightarrow D>0$

Therefore, the roots of the given equation are real and unequal.

Hence, option C is correct.