Question

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

Answer 1

Step 1: Form the equation

The given polynomial is $4s^2-4s+1$.

We know that the zeroes of a polynomial are evaluated by equating them with zero.

$\therefore p\left(s\right)=0$

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

Step 2: Solve to find the zeroes

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

$\Rightarrow {\left(2s\right)}^2-2\left(2s\right)+1^2=0$

$\Rightarrow$ ${\left(2s+1\right)}^2=0$ $\left[\because {\left(a-b\right)}^2=a^2-2ab+b^2\right]$

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

Step 3: Verification

We know that for a given polynomial $a{s}^2+bs+c$,

Sum of the zeroes$=-\frac{b}{a}$ and product of the roots$=\frac{c}{a}$.

Sum of the zeroes $=\frac{1}{2}+\frac{1}{2}=1$

Again, $-\frac{b}{a}=-\frac{\left(-4\right)}{4}=1$

Product of the zeroes $=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$

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

Thus, the relationship between the zeroes and the coefficients of the polynomial is verified.

Hence, the zeroes of this polynomial are $\frac{1}{2}$ and $\frac{1}{2}$.