Question

Given that one root is $3$ times the others for the quadratic equation $3{x^2} - 2x + p = 0$, find (a) the value of p and (b) the two roots?

Answer 1

In this question, we are dealing with quadratic equations. So, there are two equations relating the roots with constant and coefficient terms as $\alpha + \beta = \dfrac{{ - b}}{a}$ and $\alpha \beta = \dfrac{c}{a}$ for the standard equation $a{x^2} + bx + c = 0$. Also, there is a relation between the two roots in the above question. So, we can also write one root in terms of another.

Complete step-by-step answer:
In the above question, it is given that one root is $3$ times the others for the quadratic equation $3{x^2} - 2x + p = 0$.
Let one root of the quadratic equation be x.
Then the second root be $3x$.
We know that $\alpha + \beta = \dfrac{{ - b}}{a}$ and $\alpha \beta = \dfrac{c}{a}$ for the standard equation $a{x^2} + bx + c = 0$.
On comparing the standard equation with the given equation, we get
$a = 3,\,b = - 2,\,\,c = p$ and let $\alpha = x$ , $\beta = 3x$.
$\alpha + \beta = \dfrac{{ - b}}{a}$
On putting the values in above equation, we get
$x + 3x = \dfrac{{ - \left( { - 2} \right)}}{3}$
$\Rightarrow 4x = \dfrac{2}{3}$
Divide both sides by $2$.
$\Rightarrow 2x = \dfrac{1}{3}$
Now, on doing cross-multiplication, we get
$\Rightarrow x = \dfrac{1}{6}$

(a) Now, we will find the value of p in this part.
$\alpha \beta = \dfrac{c}{a}$
Now, put the values in above equation
$\left( {3x} \right)\left( x \right) = \dfrac{p}{3}$
$\Rightarrow 3{x^2} = \dfrac{p}{3}$
$\Rightarrow p = 9{x^2}$
Now, put the value of $x = \dfrac{1}{6}$ in the above equation.
$\Rightarrow p = 9{\left( {\dfrac{1}{6}} \right)^2}$
$\Rightarrow p = 9 \times \dfrac{1}{{36}}$
$\Rightarrow p = \dfrac{1}{4}$
Therefore, the value of p is $\dfrac{1}{4}$.

(b) Now, in this part we will find the value of two roots.
First root $= x = \dfrac{1}{6}$.
Second root $= \,3x = 3 \times \dfrac{1}{6} = \dfrac{1}{2}$
Therefore, the value of two roots are $\dfrac{1}{6}\,and\,\dfrac{1}{2}$.

Note: We can also find the roots of a quadratic equation using quadratic formula. From that formula we get two values of x which we can relate with the help of the condition given in the question and therefore we can find the value of p in the above question.