Question

Solve \[18{{x}^{3}}+81{{x}^{2}}+121x+60=0\] give that the root is equal to half of the sum of the remaining roots?

Answer 1

Hint: Assume that the roots of the given cubic equation are \[\alpha ,\beta \] , and \[\gamma \] . It is given that one root is equal to half the sum of the remaining roots, \[\alpha =\dfrac{\beta +\gamma }{2}\] . The given cubic equation is \[18{{x}^{3}}+81{{x}^{2}}+121x+60=0\] . Now, compare the cubic equation \[18{{x}^{3}}+81{{x}^{2}}+121x+60=0\] with the standard form of the cubic equation \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\] and get the value of \[a,b,c\] , and \[d\] . We know the formula, the sum of all roots = \[\alpha +\beta +\gamma =\dfrac{-b}{a}\] , the sum of the product of roots taken two at a time = \[\alpha \beta +\beta \gamma +\gamma \alpha =\dfrac{c}{a}\] , and the product of all roots = \[\alpha \beta \gamma =\dfrac{d}{a}\] . Now, using \[\alpha =\dfrac{\beta +\gamma }{2}\] and \[\alpha +\beta +\gamma =\dfrac{-b}{a}\] , get the value of \[\alpha \] . Now, put the value of \[\alpha \] in the equation \[\alpha =\dfrac{\beta +\gamma }{2}\] and get the value of \[\gamma \] in terms of \[\beta \] . Now, put the value of \[\alpha \] and \[\gamma \] in the equation \[\alpha \beta \gamma =\dfrac{d}{a}\] and get the value of \[\beta \] . Now, using the value of \[\beta \] gets the value of \[\gamma \] using the equation, \[-3-\beta =\gamma \] . Solve it further and get the value of \[\alpha ,\beta \] , and \[\gamma \] .

Complete step-by-step answer:
According to the question, it is given that
The given cubic equation = \[18{{x}^{3}}+81{{x}^{2}}+121x+60=0\] ……………………………………………..(1)
First of all, let us assume that the roots of the given cubic equation are \[\alpha ,\beta \] , and \[\gamma \] ………………………………..(2)
It is given that one root is equal to half the sum of the remaining roots.
\[\alpha =\dfrac{\beta +\gamma }{2}\]
\[\Rightarrow 2\alpha =\beta +\gamma \] ………………………………………..(3)
We know the standard cubic equation \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\] …………………………………………..(4)
The sum of all roots = \[\dfrac{-b}{a}\] ……………………………………(5)
The sum of the product of roots taken two at a time = \[\dfrac{c}{a}\] …………………………………………..(6)
The product of all roots = \[\dfrac{d}{a}\] ………………………………………(7)
Now, on comparing equation (1) and equation (4), we get
\[a=18\] ……………………………..(8)
\[b=81\] ………………………………..(9)
\[c=121\] …………………………………..(10)
\[d=60\] ………………………………………..(11)
From equation (5), equation (8), and equation (9), we have
The sum of all roots = \[\dfrac{-81}{18}=\dfrac{-9}{2}\] ……………………………………….(12)
We have assumed that \[\alpha ,\beta \] , and \[\gamma \] are the roots of the given cubic equation.
So, the sum of all roots = \[\alpha +\beta +\gamma \] …………………………………………..(13)
Now, from equation (3), equation (12), and equation (13), we get
\[\begin{align}
  & \Rightarrow \alpha +\left( \beta +\gamma \right)=\dfrac{-9}{2} \\
 & \Rightarrow \alpha +2\alpha =\dfrac{-9}{2} \\
 & \Rightarrow 3\alpha =\dfrac{-9}{2} \\
\end{align}\]
\[\Rightarrow \alpha =-\dfrac{3}{2}\] ………………………………..(14)
Now, from equation (3) and equation (14), we get
\[\begin{align}
  & \Rightarrow 2\times \dfrac{-3}{2}=\beta +\gamma \\
 & \Rightarrow -3=\beta +\gamma \\
\end{align}\]
\[\Rightarrow -3-\beta =\gamma \] …………………………………………..(15)
From equation (6), equation (8), and equation (11), we have
The product of all roots = \[\dfrac{-60}{18}=\dfrac{-10}{3}\] ……………………………………….(16)
We have assumed that \[\alpha ,\beta \] , and \[\gamma \] are the roots of the given cubic equation.
So, the product of all roots = \[\alpha \beta \gamma \] ……………………………………….(17)
Now, from equation (14), equation (15), equation (16), and equation (17), we get
\[\begin{align}
  & \Rightarrow \left( \dfrac{-3}{2} \right)\beta \left( -3-\beta \right)=\dfrac{-10}{3} \\
 & \Rightarrow 9\beta \left( 3+\beta \right)=-20 \\
 & \Rightarrow 9{{\beta }^{2}}+27\beta +20=0 \\
 & \Rightarrow 9{{\beta }^{2}}+15\beta +12\beta +20=0 \\
 & \Rightarrow 3\beta \left( 3\beta +5 \right)+4\left( 3\beta +5 \right)=0 \\
\end{align}\]
\[\Rightarrow \left( 3\beta +5 \right)\left( 3\beta +4 \right)=0\]
So, \[\beta =\dfrac{-5}{3}\] or \[\beta =\dfrac{-4}{3}\] …………………………………….(18)
On putting \[\beta =\dfrac{-5}{3}\] in equation (15), we get
\[\begin{align}
  & \Rightarrow -3-\left( \dfrac{-5}{3} \right)=\gamma \\
 & \Rightarrow \dfrac{-9+5}{3}=\gamma \\
\end{align}\]
\[\Rightarrow \dfrac{-4}{3}=\gamma \] …………………………………….(19)
On putting \[\beta =\dfrac{-4}{3}\] in equation (15), we get
\[\begin{align}
  & \Rightarrow -3-\left( \dfrac{-4}{3} \right)=\gamma \\
 & \Rightarrow \dfrac{-9+4}{3}=\gamma \\
\end{align}\]
\[\Rightarrow \dfrac{-5}{3}=\gamma \] …………………………………….(20)
Now, equation (14), equation (18), equation (19), and equation (20), we have the value of \[\alpha ,\beta \] , and \[\gamma \] .
\[\alpha =-\dfrac{3}{2}\] , \[\beta =\dfrac{-5}{3}\] or \[\beta =\dfrac{-4}{3}\] , and \[\gamma =\dfrac{-4}{3}\] or \[\gamma =\dfrac{-5}{3}\] .
Therefore, the roots of the given cubic equation are \[\dfrac{-3}{2}\] , \[\dfrac{-4}{3}\] , and \[\dfrac{-5}{3}\] .

Note: We can also solve this question using the factorization method.
According to the question, the given cubic equation is
\[\begin{align}
  & \Rightarrow 18{{x}^{3}}+81{{x}^{2}}+121x+60=0 \\
 & \Rightarrow 18{{x}^{3}}+27{{x}^{2}}+54{{x}^{2}}+81x+40x+60 \\
 & \Rightarrow 9{{x}^{2}}\left( 2x+3 \right)+27x\left( 2x+3 \right)+20\left( 2x+3 \right)=0 \\
 & \Rightarrow \left( 2x+3 \right)\left( 9{{x}^{2}}+27x+20 \right)=0 \\
 & \Rightarrow \left( 2x+3 \right)\left\{ \left( 9{{x}^{2}}+15x+12x+20 \right) \right\}=0 \\
 & \Rightarrow \left( 2x+3 \right)\left\{ 3x\left( 3x+5 \right)+4\left( 3x+5 \right) \right\}=0 \\
 & \Rightarrow \left( 2x+3 \right)\left( 3x+5 \right)\left( 3x+4 \right)=0 \\
\end{align}\]
Here, we have the values of x for which the value of the given cubic equation is equal to zero.
So, \[x=\dfrac{-3}{2}\] , \[x=\dfrac{-4}{3}\] , and \[x=\dfrac{-5}{3}\] .
We know the property that the roots are values of a cubic equation for which the values of the cubic equation is equal to zero.
 The roots of the given cubic equation are \[\dfrac{-3}{2}\] , \[\dfrac{-4}{3}\] , and \[\dfrac{-5}{3}\] .