Question

If \[l,m,n\] are real, and \[l \ne m\], 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) Complex
(c) Real and unequal
(d) None of these

Answer 1

Hint: Here, we need to find the nature of roots of the given quadratic equation. We will use the formula for discriminant of a quadratic equation to find the discriminant of the given equation. Then we will use the property of discriminant to find the nature of roots of the given equation.
Formula Used:
We will use the formula of the discriminant of a quadratic equation of the form \[a{x^2} + bx + c = 0\] is given by \[D = {b^2} - 4ac\].

Complete step-by-step answer:
We need to find the nature of roots of the given quadratic equation.
We will use the formula for discriminant of a quadratic equation \[a{x^2} + bx + c = 0\] to find the nature of the roots.
Comparing the equations \[\left( {l - m} \right){x^2} - 5\left( {l + m} \right)x - 2\left( {l - m} \right) = 0\] and \[a{x^2} + bx + c = 0\], we get
\[a = l - m\], \[b = - 5\left( {l + m} \right)\], and \[c = - 2\left( {l - m} \right)\]
Substituting \[a = l - m\], \[b = - 5\left( {l + m} \right)\], and \[c = - 2\left( {l - m} \right)\] in the formula \[D = {b^2} - 4ac\], we get
\[ \Rightarrow D = {\left[ { - 5\left( {l + m} \right)} \right]^2} - 4\left( {l - m} \right)\left[ { - 2\left( {l - m} \right)} \right]\]
Simplifying the expression, we get
\[ \Rightarrow D = 25{\left( {l + m} \right)^2} + 8{\left( {l - m} \right)^2}\]
We need to check whether this discriminant is more than 0, less than 0, or equal to 0.
It is given that \[l \ne m\].
Therefore, \[l - m \ne 0\].
We know that the square of any real number is always positive.
Therefore, we get
$\Rightarrow$ \[{\left( {l + m} \right)^2} > 0\]
Multiplying both sides by 25, we get
$\Rightarrow$ \[25{\left( {l + m} \right)^2} > 0\]
Since \[l - m\] is not equal to zero, the square of \[l - m\] is always positive.
Therefore, we get
$\Rightarrow$ \[{\left( {l - m} \right)^2} > 0\]
Multiplying both sides by 8, we get
$\Rightarrow$ \[8{\left( {l - m} \right)^2} > 0\]
Now, we can observe that \[25{\left( {l + m} \right)^2} > 0\] and \[8{\left( {l - m} \right)^2} > 0\].
Therefore, we can say that
$\Rightarrow$ \[25{\left( {l + m} \right)^2} + 8{\left( {l - m} \right)^2} > 0\]
Substituting \[D = 25{\left( {l + m} \right)^2} + 8{\left( {l - m} \right)^2}\] in the inequation, we get
\[ \Rightarrow D > 0\]
Thus, the discriminant of the equation $\Rightarrow$ \[\left( {l - m} \right){x^2} - 5\left( {l + m} \right)x - 2\left( {l - m} \right) = 0\] is greater than 0.
Therefore, the roots of the equation $\Rightarrow$ \[\left( {l - m} \right){x^2} - 5\left( {l + m} \right)x - 2\left( {l - m} \right) = 0\] are real and unequal.
Thus, the correct option is option (c).

Note: We used the term “quadratic equation” in our solution. A quadratic equation is an equation that has the highest degree of 2. It is of the form \[a{x^2} + bx + c = 0\], where \[a\] is not equal to 0. A quadratic equation has 2 solutions. We can find the nature of the root using the following properties of discriminant:
If \[D > 0\], then the roots of the quadratic equation are real and unequal.
If \[D = 0\], then the roots of the quadratic equation are real and equal.
If \[D < 0\], then the roots of the quadratic equation are not real, that is complex.