Question

Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficient.
$3x^2-x-4$

Answer 1

Let us first find the zeroes of the given polynomial $3x^2-x-4$ by equating it to $0$ as shown below:

$3x^2-x-4=0\Rightarrow 3x^2+3x-4x-4=0\Rightarrow 3x(x+1)-4(x+1)=0\Rightarrow (3x-4)(x+1)=0\Rightarrow (3x-4)=0,(x+1)=0\Rightarrow 3x=4,x=-1\Rightarrow x=\frac{4}{3},x=-1$

Therefore, the zeroes are $\frac{4}{3}$ and $-1$.

In the polynomial $3x^2-x-4$, we have $a=3,b=-1$ and $-4$. Now, consider

Sum of zeroes

$-1+\frac{4}{3}=\frac{-3+4}{3}=\frac{1}{3}=-\frac{(-1)}{3}=-\frac{b}{a}$

Product of zeroes

$-1\times \frac{4}{3}=-\frac{4}{3}=\frac{c}{a}$

Hence, the relation between the zeroes and coefficients is verified.