Question

If the equation ${x^2} - \left( {2 + m} \right)x + \left( { - {m^2} - 4m - 4} \right) = 0$has coincident roots, then
$
  (a){\text{ m = 0, m = 1}} \\
  (b){\text{ m = 2, m = 2}} \\
  (c){\text{ m = - 2, m = - 2}} \\
  (d){\text{ m = 6, m = 1}} \\
$

Answer 1

Hint: In this question the roots are coincident this means that the roots are equal. Form two equations using sum of the roots and product of roots , simplify them to get the values of m .

Complete step-by-step answer:

Given quadratic equation is
${x^2} - \left( {2 + m} \right)x + \left( { - {m^2} - 4m - 4} \right) = 0$
Now it is given that it has coincident roots (i.e. both the roots are the same).
So let us consider both the roots are$\alpha $.
Now as we know that in a quadratic equation the sum of roots is the ratio of negative time’s coefficient of x to the coefficient of x2.
$ \Rightarrow \alpha + \alpha = \dfrac{{ - \left( { - \left( {2 + m} \right)} \right)}}{1} = 2 + m$
$ \Rightarrow 2\alpha = 2 + m$
$ \Rightarrow \alpha = \dfrac{{2 + m}}{2}$……………………… (1)
Now we also know that in a quadratic equation the product of roots is the ratio of constant term to the coefficient of x2.
$ \Rightarrow {\alpha ^2} = \dfrac{{\left( { - {m^2} - 4m - 4} \right)}}{1} = \left( { - {m^2} - 4m - 4} \right)$…………………. (2)
Now equation (1) put the value of $\alpha $ in equation (2) we have,
$ \Rightarrow {\left( {\dfrac{{2 + m}}{2}} \right)^2} = \left( { - {m^2} - 4m - 4} \right)$
Now expand the whole square we have,
$ \Rightarrow \left( {\dfrac{{4 + {m^2} + 4m}}{4}} \right) = \left( { - {m^2} - 4m - 4} \right)$
Now simplify the above equation we have,
$\begin{gathered}
   \Rightarrow 4 + {m^2} + 4m = 4\left( { - {m^2} - 4m - 4} \right) \\
   \Rightarrow 4 + {m^2} + 4m = - 4{m^2} - 16m - 16 \\
   \Rightarrow 5{m^2} + 20m + 20 = 0 \\
\end{gathered} $
Now divide by 5 throughout we have,
$ \Rightarrow {m^2} + 4m + 4 = 0$
Now as we know this is the whole square of${\left( {m + 2} \right)^2}$.
$ \Rightarrow {\left( {m + 2} \right)^2} = 0$
$ \Rightarrow m = - 2, - 2$
So this is the required value of m if the quadratic equation has coincident roots.
Hence option (c) is correct.

Note: Whenever we face such types of problems it is always advisable to have a good grasp over the basic quadratic formula of sum or product of roots, coincident roots was a tricky word used here whose meaning is specified in hint as well. These things will help to solve problems of such kind in quadratic.