Question

If the equation ${{x}^{4}}-\left( 3m+2 \right){{x}^{2}}+{{m}^{2}}=0\left( m>0 \right)$ has four real roots which are in A.P then the value of ‘m’ is

Answer 1

We solve this problem using the standard representation of four terms of an A.P as
$\left( a-3d \right),\left( a-d \right),\left( a+d \right),\left( a+3d \right)$
Then we use the sum and product of terms of a polynomial of degree 4 that is
If $p,q,r,s$are the roots of equation $a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+e=0$ then, the sum of roots is
$\Rightarrow p+q+r+s=\dfrac{-b}{a}$
The sum of product of roots taken two at a time is
$\Rightarrow pq+qr+rs+sp+pr+sq=\dfrac{c}{a}$
The product of roots
$\Rightarrow pqrs=\dfrac{e}{a}$
By using the above formulas to given equation we find the value of $'m'$

Complete step by step answer:
We are given that the polynomial equation of degree 4 as
$\Rightarrow {{x}^{4}}-\left( 3m+2 \right){{x}^{2}}+{{m}^{2}}=0$
Let us rewrite the above equation by placing the missing terms as
$\Rightarrow {{x}^{4}}+0{{x}^{3}}-\left( 3m+2 \right){{x}^{2}}+0x+{{m}^{2}}=0$
We are given that the roots of above equation are in A.P
We know that the standard representation of four terms of an A.P as
$\left( a-3d \right),\left( a-d \right),\left( a+d \right),\left( a+3d \right)$
Let us assume that the roots of given equation as
$\left( a-3d \right),\left( a-d \right),\left( a+d \right),\left( a+3d \right)$
We know that the sum of roots and product of roots of equation of degree 4 that is
If $p,q,r,s$are the roots of equation $a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+e=0$ then, the sum of roots is
$\Rightarrow p+q+r+s=\dfrac{-b}{a}$
The sum of product of roots taken two at a time is
$\Rightarrow pq+qr+rs+sp+pr+sq=\dfrac{c}{a}$
The product of roots
$\Rightarrow pqrs=\dfrac{e}{a}$
By using the above formulas to given equation we find the value of $'m'$
Now, by using the sum of roots to given equation we get

& \Rightarrow a-3d+a-d+a+d+a+3d=\dfrac{-0}{1}=0 \\\ & \Rightarrow 4a=0 \\\ & \Rightarrow a=0 \\\ \end{aligned}$$ Now the roots of given equation will be $$-3d,-d,d,3d$$ Now by using the sum of product of roots taken two at a time we get $$\begin{aligned} & \Rightarrow \left( -3d\times -d \right)+\left( -d\times d \right)+\left( d\times 3d \right)+\left( 3d\times -3d \right)+\left( -3d\times d \right)+\left( -d\times 3d \right)=\dfrac{-\left( 3m+2 \right)}{1} \\\ & \Rightarrow -10{{d}^{2}}=-\left( 3m+2 \right) \\\ & \Rightarrow {{d}^{2}}=\dfrac{\left( 3m+2 \right)}{10} \\\ \end{aligned}$$ Now by using the product of roots formula we get $$\begin{aligned} & \Rightarrow -3d\times -d\times d\times 3d=\dfrac{{{m}^{2}}}{1} \\\ & \Rightarrow 9{{\left( {{d}^{2}} \right)}^{2}}={{m}^{2}} \\\ \end{aligned}$$ Now, by substituting the value of $$'{{d}^{2}}'$$ in above equation we get $$\begin{aligned} & \Rightarrow 9{{\left( \dfrac{3m+2}{10} \right)}^{2}}={{m}^{2}} \\\ & \Rightarrow 9\left( 9{{m}^{2}}+12m+4 \right)=100{{m}^{2}} \\\ & \Rightarrow 19{{m}^{2}}-108m-36=0 \\\ \end{aligned}$$ Now by using the factorisation method that is rewriting the middle in such a way that we get factors then we get $$\begin{aligned} & \Rightarrow 19{{m}^{2}}-114m+6m-36=0 \\\ & \Rightarrow 19m\left( m-6 \right)+6\left( m-6 \right)=0 \\\ & \Rightarrow \left( m-6 \right)\left( 19m+6 \right)=0 \\\ \end{aligned}$$ We know that if $$a\times b=0$$ then either of $$a,b$$ will be zero. By using this result to above equation we get the first term as $$\Rightarrow m=6$$ Similarly, by taking the second term we get $$\Rightarrow m=\dfrac{-6}{19}$$ **We are given that $$m>0$$ so, the value of $$'m'$$ is 6.** **Note:** Students may make mistakes in taking the terms of an A.P. The general representation of A.P is $$a,\left( a+d \right),\left( a+2d \right),\left( a+3d \right),.......$$ Students may take these four terms which gives the complex solution. But if we consider the terms as $$\left( a-3d \right),\left( a-d \right),\left( a+d \right),\left( a+3d \right)$$ Here, we get the solution easily because while adding the terms we get only one variable remains which will be easy to solve. This point needs to be taken care of.