Question

Find the zeroes of the following quadratic polynomials and verify the relationship between zeroes and their coefficients.

$x^2-5x+6$

Answer 1

Finding the zeroes:

$p\left(x\right)=x^2-5x+6$

$$\begin{gathered} p\left(x\right)=x^2-2x-3x+6 \\ p\left(x\right)=x(x-2)-3(x-2) \\ p\left(x\right)=(x-3)(x-2) \end{gathered}$$

Therefore, $$\begin{gathered} (x-3)=0 \\ x=3 \end{gathered}$$or $$\begin{gathered} (x-2)=0 \\ x=2 \end{gathered}$$

The two zeroes of the polynomial are $3and2$

Verifying the relationship between the zeroes and their coefficients.

Let the zeroes be $\alpha and\beta$

$\alpha =3and\beta =2$

Sum of zeroes=$\alpha +\beta =-\frac{b}{a}$$=\frac{-coefficientofx}{coefficientof{x}^2}$$$\begin{gathered} =\frac{-(-5)}{1} \\ =5 \end{gathered}$$

$Sumofzeroes=3+2=5$

Product of zeroes$=\alpha \beta =\frac{c}{a}=\frac{cons\tan tterm}{coefficientof{x}^2}=\frac{6}{1}$

Product of zeroes$=3\times 2=6$

Hence, zeroes of the polynomial are 2,3 and the relation is verified.