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# 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

Last updated date: 20th Jun 2024
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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 $D = {b^2} - 4ac$.

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.