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If one root of the polynomial \[k{{x}^{2}}-15x+18=0\] is 2, then find the value of k?

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Last updated date: 23rd Jul 2024
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Answer
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Hint:
We are given that x = 2 is the root of the given equation \[k{{x}^{2}}-15x+18=0\]. So, x = 2 will satisfy this equation. Substitute x = 2 in the equation and find the value of k.

Complete step-by-step answer:
Here, we are given that one of the roots of the polynomial \[k{{x}^{2}}-15x+18=0\] is 2 then we have to find the value of k.
First of all, we must know that roots or zeros of the given equation satisfy it or in other words, we can say that by substituting the value of roots in place of a variable in any equation, the equation becomes zero. Here variable is x, in terms of which that equation of polynomial is written.
Let us consider the polynomial given in this question.
\[k{{x}^{2}}-15x+18=0....\left( i \right)\]
We are given that 2 is the root of this polynomial. This means that x = 2 will satisfy this polynomial.
By substituting x = 2 in equation (i), we get,
\[k{{\left( 2 \right)}^{2}}-15\left( 2 \right)+18=0\]
By simplifying the above equation, we get,
\[\begin{align}
 & 4k-30+18=0 \\
 & \Rightarrow 4k-12=0 \\
\end{align}\]
By adding 12 on both sides of the above equation, we get,
\[4k-12+12=12\]
Or, \[4k=12\]
By dividing 4 on both sides of the equation, we get,
\[\dfrac{4k}{4}=\dfrac{12k}{4}\]
Or \[k=\dfrac{12}{4}\]
By simplifying the RHS of the above equation, we get,
\[\Rightarrow k=3\]
So, we get the value of k equal to 3.

Note:
In this question, we can cross-check our answer as follows:
Let us consider the polynomial \[k{{x}^{2}}-15x+18=0\]
By substituting k = 3 and x = 2 in the above polynomial, we get,
\[\left( 3 \right){{\left( 2 \right)}^{2}}-15\left( 2 \right)+18=0\]
By simplifying the above equation, we get,
\[3\left( 4 \right)-30+18=0\]
\[\Rightarrow 12-30+18=0\]
\[\Rightarrow 0=0\]
LHS = RHS
Since, LHS = RHS, therefore our answer is correct.