Question

Find the value of $p$; if one root of the quadratic equation $3x^2-px-6=0$ is $3$. Also, find the second (other) root of the equation.

Answer 1

Step 1: Given information in the question

The given quadratic equation is $3x^2-px-6=0$.

On comparing it with a general quadratic equation we get:

$a=3,b=-p,c=-6$

One root of the quadratic equation is $3$.

Let the other root of the quadratic equation be $x$.

Step 2: Apply the condition of the sum of roots of the quadratic equation.

$\begin{array}{rcl}\text{Sum of the roots}& =& \frac{-b}{a}\\ x+3& =& \frac{-b}{a}\\ x+3& =& \frac{-(-p)}{3}\\ 3x+9& =& p.....\left(1\right)\end{array}$

Step 3: Apply the condition of the product of roots of the quadratic equation.

$\begin{array}{rcl}\text{Product of the roots}& =& \frac{c}{a}\\ 3x& =& \frac{c}{a}\\ 3x& =& \frac{-6}{3}\\ 3x& =& -2.....\left(2\right)\\ x& =& \frac{-2}{3}\end{array}$

Substitute equation (2) in equation (1) to get:

$\begin{array}{rcl}3x+9& =& p\\ -2+9& =& p\\ p& =& 7\end{array}$

Hence, the value of $p$ is 7, and the value of the other root is $\frac{-2}{3}$.