Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
$4s^2-4s+1$

Answer 1

Let $f\left(s\right)=4s^2-4s+1$

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

$4s^2-4s+1=0$

$4s^2-2s-2s+1=0$

$2s\left(2s-1\right)-1\left(2s-1\right)=0$

$\left(2s-1\right)\left(2s-1\right)=0$

$s=\frac{1}{2},s=\frac{1}{2}$

The zeros of the given equation is $\frac{1}{2}$ and $\frac{1}{2}$.

Sum of the zeros is $\frac{1}{2}+\frac{1}{2}=1$.

Product of the zeros is $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$.

According to the given equation,

The sum of the zeros is,

$\frac{-b}{a}=\frac{-\left(-4\right)}{4}$

$=1$

The product of the zeros is,

$\frac{c}{a}=\frac{1}{4}$

Hence, it is verified that,

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

And,

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