Question

$\text{Find the zeroes of the polynomial}p\left(x\right)=4x^2-4x+1\text{and verify the relationship between the zeroes and the coefficients.}$

Answer 1

$$\begin{gathered} \mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{}\mathbf{1}\mathbf{:}\mathbf{}\mathbf{\text{Find the zeroes:}} \\ \text{Here, the polynomial is}p\left(x\right)=4x^2-4x+1 \\ \text{put}p\left(t\right)=0 \\ \begin{array}{cc}\Rightarrow & 4t^2-4t+1=0\\ \Rightarrow & (2t-1)(2t-1)=0\\ \Rightarrow & t=\frac{1}{2},\frac{1}{2}\end{array} \\ \text{Therefore,}\alpha =\frac{1}{2}=\beta \\ \mathbf{\text{Step 2: Verify the relation between zeroes and coefficients:}} \\ \text{Here,}a=4;b=-4;c=1 \\ \begin{array}{cc}\alpha +\beta & =\frac{1}{2}+\frac{1}{2}\\ & =1\\ & =\frac{4}{4}\\ & =\frac{-b}{a}\end{array} \\ \begin{array}{cc}\alpha \beta & =\frac{1}{2}\times \frac{1}{2}\\ & =\frac{1}{4}\\ & =\frac{c}{a}\end{array} \end{gathered}$$

Thus, the relationship between zeroes and coefficients are verified.