Question

If one of the roots of equation $x^2+ax+3=0$ is $3$ and one of the roots of the equation $x^2+ax+b=0$ is three times the other root, then the value of $b$ is

• A. $3$
• B. $4$
• C. $2$
• D. $1$

Answer 1

The correct option is A

$3$

Explanation for the correct option.

Step 1: Prerequisites for the solution.

The quadratic equation is given as $x^2+ax+b=0$ then the coefficient of $x$ that is $a$ is equal to the sum of its roots and the term $b$ is equal to the product of its roots.

So, suppose that the roots of the above equation are $c\text{and}d$ then $c+d=a$ and $cd=b$.

Step 2: Calculation for the value of $\mathit{a}$,

Here, the equation $x^2+ax+3=0$ has one of its roots as $3$. Let the other root be $e$ then we will get two equations that are,

$3+e=-a\dots \left(1\right)$

Here $a$ will be negative because the equation will not have real roots otherwise.

$3e=3\dots \left(2\right)$

From equation $\left(2\right)$, we have

$e=1$

Now, from the equation $\left(1\right)$, we have

$a=-4$

Step 3: Calculation for the value of $\mathit{b}$.

We know that the value of $a$ is $-4$ then the second equation will become $x^2-4x+b=0$.

It is given that one root of the equation $x^2-4x+b=0$ is three times the other. Suppose that one root is $f$ then the other root will be $3f$.

Now, the sum of the roots is equal to $-4$,

$\begin{array}{rcl}\therefore f+3f& =& -4\\ & \Rightarrow & f=-1\end{array}$

The other root will be $-3$.

Finally $b$ is equal to the product of these roots,

$\begin{array}{rcl}\therefore b& =& -3\times -1\\ & =& 3\end{array}$

Hence, the correct option is (A).