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If one root of the equation, \[3{{x}^{2}}-9x=kx-k\] is 2, then the value of k is: -
(a) 4
(b) 3
(c) -6
(d) -8

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Last updated date: 17th Jun 2024
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Answer
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Hint: First of all take all the terms of the given quadratic equation to the left so that we can get 0 in the right hand side. Now, assume the two roots of the given quadratic equation as \[\alpha \] and \[\beta \] with \[\alpha =2\]. Apply the identity: - \[\alpha +\beta =\dfrac{-b}{a}\] and \[\alpha \beta =\dfrac{c}{a}\] to form two linear equation in k and \[\beta \] and find the value of k. Remember that b = co-efficient of x and c = constant term while a = co – efficient of \[{{x}^{2}}\].

Complete step-by-step solution
We have been given the quadratic equation: -
\[\Rightarrow \]\[3{{x}^{2}}-9x=kx-k\]
Taking all the terms to the L.H.S, we get,
\[\begin{align}
  & \Rightarrow 3{{x}^{2}}-9x-kx+k=0 \\
 & \Rightarrow 3{{x}^{2}}-x\left( 9+k \right)+k=0 \\
\end{align}\]
Let us assume the two roots of the above equation as \[\alpha \] and \[\beta \]. We have provided with one root equal to 2. Let \[\alpha =2\]. Here, we are assuming the coefficient of \[{{x}^{2}}\], x, and the constant term is denoted by a, b and c respectively.
\[\Rightarrow a=3,b=-\left( 9+k \right)\] and $c = k.$
Now, we know that sum of the roots of a quadratic equation is the ratio of (-b) and (a) and product of the roots is the ratio of (c) and (a).
\[\begin{align}
  & \Rightarrow \alpha +\beta =\dfrac{-b}{a} \\
 & \Rightarrow 2+\beta =\dfrac{9+k}{3} \\
 & \Rightarrow 6+3\beta =9+k \\
\end{align}\]
\[\Rightarrow 3\beta -k=3\] ----- (i)
Also, we have,
\[\begin{align}
  & \Rightarrow \alpha \beta =\dfrac{c}{a} \\
 & \Rightarrow 2\beta =\dfrac{k}{3} \\
\end{align}\]
\[\Rightarrow \beta =\dfrac{k}{6}\] ------ (ii)
Substituting the value of \[\beta \] from equation (ii) in equation (i), we have,
\[\begin{align}
  & \Rightarrow 3\times \dfrac{k}{6}-k=3 \\
 & \Rightarrow \dfrac{k}{2}-k=3 \\
 & \Rightarrow -\dfrac{k}{2}=3 \\
 & \Rightarrow k=-6 \\
\end{align}\]
Hence, option (c) is the correct answer.

Note: You may note that we have assumed \[\alpha =2\]. You may assume, \[\beta =2\], this will not affect the value of k in the answer. The only difference it will make is that at last, we have to eliminate \[\alpha \] instead of \[\beta \] just like we did above. There can be another way to solve the question if options are given. We will substitute the value of k one – by – one and find the roots. The option which will give one of the roots as 2 will be the answer.