Question

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p\left(x\right)=a{x}^2+bx+c,(a\ne 0)$ then _________ $=\frac{c}{a}$

• A. $\alpha +\beta$
• B. $\alpha -\beta$
• C. $\alpha .\beta$
• D. $\frac{\alpha }{\beta }$

Answer 1

The correct option is C $\alpha .\beta$

Given that

$p\left(x\right)=a{x}^2+bx+c$ .........(1) where $a\ne 0$ is a quadratic polynomial

And $\alpha ,\beta$ are the zeroes of the polynomial $p\left(x\right)$

We can also write the polynomial in this way

$p\left(x\right)=(x-\alpha )(x-\beta )$

$\Rightarrow p\left(x\right)=x^2-(\alpha +\beta )x+\alpha \beta$ .............(2)

Equation (1) and (2) are same

Comparing the coefficients

$\frac{a}{1}=\frac{b}{-\alpha -\beta }=\frac{c}{\alpha \beta }$

On solving we get the following results

$\alpha +\beta =\frac{-b}{a}$

$\alpha .\beta =\frac{c}{a}$

Hence, option $C$ is correct.