Question

If the equation ${x^2} - (2 + m)x + 1({m^2} - 4m + 4) = 0$ has coincident roots, then:
A) $m = 0$
B) $m = 6$
C) $m = 2$
D) $m = \dfrac{2}{3}$
(This question has multiple correct answers)

Answer 1

Hint: We will first compare this to the general quadratic equation and then understand what discriminant we get when we have coincident roots. Then, we will just get the answer.

Complete step-by-step solution:
The general quadratic equation is given by $a{x^2} + bx + c = 0$. It’s discriminant is given by $D = {b^2} - 4ac$.
Now, if we compare the equation given to us by this equation, we will get:-
$ \Rightarrow a = 1,b = - (2 + m)$ and $c = {m^2} - 4m + 4$ …………….(1)
Now, let us understand what discriminant says about the roots of quadratic equations.
If D > 0, then we have real and distinct roots of the equation, if D < 0, then we have imaginary roots of the equation and if D = 0, we get real and coincident roots.
Since we are given that ${x^2} - (2 + m)x + 1({m^2} - 4m + 4) = 0$ has coincident roots, therefore, its discriminant must be equal to 0. ………….(2)
Since, $D = {b^2} - 4ac$ and we will put in these coefficient using equation (1), we will get:-
\[ \Rightarrow D = {\left\{ { - \left( {2 + m} \right)} \right\}^2} - 4 \times 1 \times \left( {{m^2} - 4m + 4} \right)\]
Simplifying the RHS a bit to get the following expression:-
\[ \Rightarrow D = {\left( {2 + m} \right)^2} - 4\left( {{m^2} - 4m + 4} \right)\]
Simplifying the RHS further using ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$to get the following expression:-
\[ \Rightarrow D = 4 + {m^2} + 4m - 4{m^2} + 16m - 16\]
Combining the like terms in the RHS to get the following expression:-
\[ \Rightarrow D = - 12 - 3{m^2} + 20m\]
Using (2) in the above expression, we will get:
\[ \Rightarrow - 3{m^2} + 20m - 12 = 0\]
We can rewrite this expression as:-
\[ \Rightarrow - 3{m^2} + 2m + 18m - 12 = 0\]
\[ \Rightarrow m\left( { - 3m + 2} \right) - 6\left( { - 3m + 2} \right) = 0\]
\[ \Rightarrow \left( { - 3m + 2} \right)\left( {m - 6} \right) = 0\]
$\therefore $ either $m = 6$ or $m = \dfrac{2}{3}$.

$\therefore $ The correct options are (B) and (D).

Note: The students must commit to memory that:-
If D > 0, then we have real and distinct roots of the equation, if D < 0, then we have imaginary roots of the equation and if D = 0, we get real and coincident roots.
The students must know that $D = {b^2} - 4ac$ is known as discriminant because it discriminates between the possible types of answers as we just stated above.
We always get a parabolic form of curve with quadratic equations.