Question

If the product of the zeroes of the polynomial $ a{x^2} - 5x + 6 $ is $ 5 $ then find the value of $ a $ .

Answer 1

Hint: In this type of question, the quadratic polynomial concept will be used. As we know that for a quadratic polynomial which is of the form $ a{x^2} + bx + c $ then the product of the zeroes will be equal to $ \dfrac{c}{a} $ . So on substituting the values, and solving we will get the value for $ a $ .
Formula used:
A quadratic polynomial of the form $ a{x^2} + bx + c $ , then
The sum of the zeroes will be given by $ \dfrac{{ - b}}{a} $
And the product of the zeroes will be given by $ \dfrac{c}{a} $
Here,
 $ a,b\& c $ , will be the variables.

Complete step-by-step answer:
So we have the equation given $ a{x^2} - 5x + 6 $ . So from this firstly we will identify the values of $ a,b\& c $ .
So on comparing the equation by the quadratic polynomial equation which is given by $ a{x^2} + bx + c $ , we get
 $ \Rightarrow b = - 5{\text{ , c = 6}} $
From the formula, we already know that the product of the zeroes will be given by $ \dfrac{c}{a} $ and in the question, it is given that it is equal to $ 5 $
So from the above statement on framing the equation, we get the equation as
 \[ \Rightarrow \dfrac{c}{a} = 5\]
Now on substituting the values of known terms, we get
 \[ \Rightarrow \dfrac{6}{a} = 5\]
Now on doing the cross multiplication, we will have the equation as
 $ \Rightarrow 5a = 6 $
And on solving for the value of $ a $ , we get
 $ \Rightarrow \dfrac{6}{5} $
Hence, the value of $ a $ will be $ \dfrac{6}{5} $
So, the correct answer is “ $ \dfrac{6}{5} $ ”.

Note: So while solving this type of question we have to take care of one thing which is the sign mistake. As we can see the quadratic polynomial equation is positive and if we have any equation having any term negative so while assigning the values in place of it we also take the sign from it. Rest is just by memorizing the formula we can easily solve it.