Question

The sum of the roots and the product of the roots of a quadratic equation \[3{{x}^{2}}-\left( 2K+1 \right)x-K-5=0\] are equal. The value of \[K\] will be
(a) \[-\dfrac{1}{2}\]
(b) -2
(c) \[\dfrac{1}{2}\]
(d) 5

Answer 1

Hint: In this type of question we have to use the concept of finding roots of the quadratic equation if the sum and product of the roots are known. We know that if we have a quadratic equation \[a{{x}^{2}}+bx+c=0\] then the sum of its roots is given by \[-\dfrac{b}{a}\] and the product of roots is given by \[\dfrac{c}{a}\].

Complete step-by-step solution:
Now, we have to find the roots of the quadratic equation \[3{{x}^{2}}-\left( 2K+1 \right)x-K-5=0\] where we have given that the sum of the roots and product of the roots are equal.
Now, we rewrite the given quadratic equation as \[3{{x}^{2}}-\left( 2K+1 \right)x-\left( K+5 \right)=0\]
Let us compare the given equation with \[a{{x}^{2}}+bx+c=0\] we get,
\[\Rightarrow a=3,b=-\left( 2K+1 \right),c=-\left( K+5 \right)\]
We know that if we have a quadratic equation \[a{{x}^{2}}+bx+c=0\] then the sum of its roots is given by \[-\dfrac{b}{a}\] and the product of roots is given by \[\dfrac{c}{a}\]. Also we have given that for the given quadratic equation sum of the roots and product of the roots are equal. Hence, we can write
\[\begin{align}
  & \Rightarrow \text{Sum of the roots = Product of the roots} \\
 & \Rightarrow -\dfrac{b}{a}=\dfrac{c}{a} \\
\end{align}\]
By comparing the fractions, we get,
\[\Rightarrow -b=c\]
Let us substitute the values of b and c we get
\[\begin{align}
  & \Rightarrow -\left[ -\left( 2K+1 \right) \right]=-\left( K+5 \right) \\
 & \Rightarrow 2K+1=-K-5 \\
\end{align}\]
By simplification we can write,
\[\begin{align}
  & \Rightarrow 2K+K=-5-1 \\
 & \Rightarrow 3K=-6 \\
 & \Rightarrow K=-2 \\
\end{align}\]
Hence, if the sum of the roots and the product of the roots of a quadratic equation \[3{{x}^{2}}-\left( 2K+1 \right)x-K-5=0\] are equal then the value of \[K\] will be \[-2\].
Hence, option (b) is the correct option.

Note: In this type of question students have to remember the formulas for sum and product of the roots of a quadratic equation. Also students have to note that the two fractions are equal if their numerators as well as denominators are equal.