Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
$3x^2-x-4$

Answer 1

Let $f\left(x\right)=3x^2-x-4$

To calculate the zeros of the given equation, put $f\left(x\right)=0$.

$3x^2-x-4=0$

$3x^2-4x+3x-4=0$

$x\left(3x-4\right)+1\left(3x-4\right)=0$

$\left(x+1\right)\left(3x-4\right)=0$

$x=-1,x=\frac{4}{3}$

The zeros of the given equation is $-1$ and $\frac{4}{3}$.

Sum of the zeros is $-1+\frac{4}{3}=\frac{1}{3}$.

Product of the zeros is $-1\times \frac{4}{3}=-\frac{4}{3}$.

According to the given equation,

The sum of the zeros is,

$\frac{-b}{a}=\frac{-\left(-1\right)}{3}$

$=\frac{1}{3}$

The product of the zeros is,

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

Hence, it is verified that,

$sumofzeros=\frac{-coefficientofx}{coefficientof{x}^2}$

And,

$productofzeros=\frac{constantterm}{coefficientof{x}^2}$