Question

Find \[m\] so that roots of the equation \[\left( {4 + m} \right){x^2} + \left( {m + 1} \right)x + 1 = 0\] may be equal.

Answer 1

Hint: Here we have to find the value of the variable \[m\]. The given quadratic equation has equal roots, so the discriminant of the given equation is zero. Thus, we will calculate the discriminant and equate it with zero. From there, we will get the equation including \[m\] as a variable. After solving the obtained equation, we get the required value of the variable \[m\].

Formula Used: We will use the formula of discriminant \[D = {b^2} - 4ac\], where \[b\] is the coefficient of\[x\], \[c\] is the constant term of equation and \[a\] is the coefficient of \[{x^{}}\].

Complete step-by-step answer:
It is also given that the given equation has equal roots. So the discriminant of the given quadratic equation is equal to zero.
\[D = {\left( {m + 1} \right)^2} - 4 \times \left( {4 + m} \right) \times 1\] Using the formula \[D = {b^2} - 4ac\] to get the discriminant of \[\left( {4 + m} \right){x^2} + \left( {m + 1} \right)x + 1 = 0\], we get
When the roots of a given equation are equal, then the discriminant is equal to zero. So, equating the discriminant with zero, we get
\[ \Rightarrow {\left( {m + 1} \right)^2} - 4 \times \left( {4 + m} \right) \times 1 = 0\]
On simplifying the terms, we get
\[ \Rightarrow {m^2} + 1 + 2m - 4m - 16 = 0\]
On further simplification, we get
\[ \Rightarrow {m^2} - 2m - 15 = 0\]
On factoring the equation, we get
\[\begin{array}{l} \Rightarrow {m^2} - 5m + 3m - 15 = 0\\ \Rightarrow \left( {m - 5} \right)m + 3\left( {m - 5} \right) = 0\end{array}\]
In further simplification, we get
\[ \Rightarrow \left( {m - 5} \right)\left( {m + 3} \right) = 0\]
Applying the zero product property, we get
\[\begin{array}{l} \Rightarrow m - 5 = 0\\ \Rightarrow m = 5\end{array}\]
Or
\[\begin{array}{l} \Rightarrow m + 3 = 0\\ \Rightarrow m = - 3\end{array}\]
Thus, the possible values of \[m\] are 5 and \[ - 3\].

Note: There are two roots of any quadratic equation, and similarly the roots of the cubic equation are three and so on. Generally, the number of possible roots of any equation is equal to the highest power of the equation. The values of the roots always satisfy their respective equation i.e. when we put values of the roots in their respective equation; we get the value as zero.