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Find the equation of \[k\] for which the quadratic equation \[\left( {3k + 1} \right){x^2} + 2\left( {k + 1} \right)x + 1 = 0\] has real and equal roots.

Last updated date: 20th Jun 2024
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Here we have to find the value of the variable \[k\]. We will find the discriminant of the given quadratic equation. As the roots are real and equal, we will equate the value of the discriminant to 0. Then we will get the equation including \[k\] as a variable. After solving the obtained equation, we will get the required value of the variable \[k\].

Formula used:
We will use the formula of the 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 solution:
We will first find the value of the discriminant of the given equation.
Substituting the value of \[a\], \[b\] and \[c\] from the given quadratic equation in the formula of discriminant \[D = {b^2} - 4ac\] , we get
\[ \Rightarrow D = {\left[ {2\left( {k + 1} \right)} \right]^2} - 4 \times \left( {3k + 1} \right) \times 1\]
Now, we know that the roots of the given equation are real and equal. Therefore, the discriminant will be 0.
Equating the above equation to 0, we get
\[ \Rightarrow {\left[ {2\left( {k + 1} \right)} \right]^2} - 4 \times \left( {3k + 1} \right) \times 1 = 0\]
On simplifying the terms, we get
\[ \Rightarrow 4{k^2} + 4 + 8k - 12k - 4 = 0\]
Adding the like terms, we get
\[ \Rightarrow 4{k^2} - 4k = 0\]
On factoring the equation, we get
\[ \Rightarrow 4k\left( {k - 1} \right) = 0\]
Now applying zero product property, we can write
\[\begin{array}{l} \Rightarrow 4k = 0\\ \Rightarrow k = 0\end{array}\]
\[\begin{array}{l} \Rightarrow \left( {k - 1} \right) = 0\\ \Rightarrow k = 1\end{array}\] .
Thus, the possible values of \[k\] are 0 and 1.

We need to keep in mind that the roots of any quadratic equation are two, and similarly the roots of the cubic equation are three and so on. Generally, the numbers of possible roots of any equation are 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.