Question

The difference between two roots of the equation $x^3-13x^2+15x+189=0$ is $2$. Then, the roots of the equation are

• A. $-3,7,9$
• B. $-3,–7,–9$
• C. $-3,–5,7$
• D. $-3,-7,9$

Answer 1

The correct option is A

$-3,7,9$

Explanation for the correct option :

Step 1 : Constructing a system of three equations

Given : Difference between two roots of $x^3-13x^2+15x+189=0$ is $2$

Let $\alpha ,\alpha +2$ and $\beta$ be the roots of the given equation $x^3-13x^2+15x+189=0$.

Formula to be used : We know that if $p,q,r$ be the roots of the equation $x^3+a{x}^2+bx+c=0$, then,

$p+q+r=-a\left(i\right)$

$pq+qr+pr=b\left(ii\right)$

$pqr=-c\left(iii\right)$.

Here, the given equation is $x^3-13x^2+15x+189=0$ and the roots are $\alpha ,\alpha +2$ and $\beta$. So, we get the following three equation:

Applying the values in equation $\left(i\right)$

$$\begin{gathered} \Rightarrow \alpha +(\alpha +2)+\beta =13 \\ \Rightarrow 2\alpha +\beta =11\left(iv\right) \end{gathered}$$

Applying the values in equation $\left(ii\right)$

$$\begin{gathered} \Rightarrow \alpha (\alpha +2)+(\alpha +2)\beta +\alpha \beta =15 \\ \Rightarrow {\alpha }^2+2\alpha \beta +2\alpha +2\beta =15\left(v\right) \end{gathered}$$

Applying the values in equation $\left(iii\right)$

$$\begin{gathered} \Rightarrow \alpha (\alpha +2)\beta =-189 \\ \Rightarrow {\alpha }^2\beta +2\alpha \beta =-189 \end{gathered}$$

Step 2 : Solving the equations using the substitution method

From the equation $\left(iv\right)$, we get $\beta =11-2\alpha$. Substituting this in equation $\left(v\right)$, we get :

$$\begin{gathered} \Rightarrow {\alpha }^2+2\alpha (11-2\alpha )+2\alpha +2(11-2\alpha )=15 \\ \Rightarrow {\alpha }^2+22\alpha -4{\alpha }^2+2\alpha +22-4\alpha =15 \\ \Rightarrow -3{\alpha }^2+20\alpha +7=0 \\ \Rightarrow 3{\alpha }^2-20\alpha -7=0 \\ \Rightarrow 3{\alpha }^2-21\alpha +\alpha -7=0 \\ \Rightarrow 3\alpha (\alpha -7)+1(\alpha -7)=0 \\ \Rightarrow (\alpha -7)(3\alpha +1)=0 \end{gathered}$$

This implies either $\alpha =-\frac{1}{3}$ or $\alpha =7$.

Step 3: Finding the roots

When $\alpha =-\frac{1}{3}$,

$$\begin{gathered} \alpha +2=-\frac{1}{3}+2 \\ =\frac{5}{3} \end{gathered}$$

$$\begin{gathered} \beta =11-2\alpha \\ =11-2\frac{1}{3} \\ =\frac{31}{3} \end{gathered}$$

Then the three roots will be $-\frac{1}{3},\frac{5}{3},\frac{31}{3}$.

When $\alpha =7$,

$$\begin{gathered} \alpha +2=7+2 \\ =9 \end{gathered}$$

and

$$\begin{gathered} \beta =11-2\alpha \\ =11-14 \\ =-3 \end{gathered}$$

Then, the three roots will be $7,9,-3$.

Hence, option (A) is the correct answer.