Question

If the sum of the roots of the equation ${{x}^{2}}+px+q=0$ is 3 times their difference, then
A. $2{{p}^{2}}=9q$
B. $2{{p}^{2}}=9p$
C. ${{(p+q)}^{2}}=8pq$
D. None

Answer 1

Hint: In this question, we are to find the relation between the roots of the given quadratic equation. For this, we use the sum and product of the roots of a quadratic equation formula. By simply applying the coefficients of the given equation into the formulae, we get the required expression.



Formula UsedConsider a quadratic equation $a{{x}^{2}}+bx+c=0$.
Then, its roots are $\alpha $ and $\beta $. So, the sum of the roots and the product of the roots is formulated as:
$\alpha +\beta =\dfrac{-b}{a};\alpha \beta =\dfrac{c}{a}$




Complete step by step solution:The given quadratic equation is ${{x}^{2}}+px+q=0$ and its roots are $\alpha $ and $\beta $.
So, comparing the given equation with the standard equation, we get
$a=1;b=p;c=q$
Then, the sum of the roots of the given equation is obtained as
$\alpha +\beta =\dfrac{-p}{1}=-p\text{ }...(1)$
And the product of the roots of the given equation is
$\alpha \beta =\dfrac{q}{1}=q\text{ }...(2)$
It is given that the sum of the roots of the equation is 3 times their difference. Thus, we can write
$(\alpha +\beta )=3(\alpha -\beta )$
On simplifying, we get
$\begin{align}
  & \Rightarrow \alpha +\beta =3(\alpha -\beta ) \\
 & \Rightarrow \alpha +\beta =3\alpha -3\beta \\
 & \Rightarrow 2\alpha =4\beta \\
 & \Rightarrow \alpha =2\beta \text{ }...(3) \\
\end{align}$
On substituting (3) in (1), we get
$\begin{align}
  & \alpha +\beta =-p \\
 & \Rightarrow 2\beta +\beta =-p \\
 & \Rightarrow 3\beta =-p \\
 & \Rightarrow \beta =\dfrac{-p}{3}\text{ }...(4) \\
\end{align}$
On substituting (3) in (2), we get
$\begin{align}
  & \alpha \beta =q \\
 & \Rightarrow (2\beta )\beta =q \\
 & \Rightarrow 2{{\beta }^{2}}=q\text{ }...(5) \\
\end{align}$
Then, substituting (4) in (5), we get
$\begin{align}
  & 2{{\beta }^{2}}=q \\
 & \Rightarrow 2{{\left( \dfrac{-p}{3} \right)}^{2}}=q \\
 & \Rightarrow \dfrac{2{{p}^{2}}}{9}=q \\
 & \therefore 2{{p}^{2}}=9q \\
\end{align}$
Thus, this is the required expression.


Option A is correct

Note: Here, we need to remember the sum and product formulae. So, we can substitute these values in the given relation to find an expression. And be sure with the signs in the formulae. Otherwise, the entire calculation may go in the wrong way.