Find the quadratic polynomial, sum and product of whose zeroes are –1 and – 20 respectively. Also find the zeroes of the polynomial so obtained. <div style="float:le

Find the quadratic polynomial, sum and product of whose zeroes are –1 and – 20 respectively. Also find the zeroes of the polynomial so obtained.

Answers (1)

Given Sum of roots, $\alpha +\beta = -1---(i)$
Product of roots, $\alpha \beta = -20---(ii)$
We know that in $ax^{2}+bx+c= 0$
$\alpha +\beta = \frac{-b}{a}---(iii)$
$\alpha \beta = \frac{c}{a}---(iv)$
$\Rightarrow$ compare (i) & (ii) And (ii) and (iv)
$\Rightarrow \frac{-b}{a}= -1\Rightarrow b= a$
$\frac{c}{a}= -20\Rightarrow c= -20a$
In polynomial $ax^{2}+bx+c= 0$ , put values of b and c
$\Rightarrow ax^{2}+ax-20a= 0$
$\Rightarrow x^{2}+x-20= 0$
Now to obtain Zeroes of this equation apply Shri dharacharye's formula
$\left ( \alpha ,\beta \right )= \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$
$\left ( \alpha ,\beta \right )= \frac{-1\pm \sqrt{1-4\times 1\times \left ( -20 \right )}}{2\times 1}$
$= \frac{-1\pm \sqrt{81}}{2}= \frac{-1\pm 9}{2}$
$\left ( \alpha ,\beta \right )= \left ( \frac{-1+9}{2},\frac{-1-9}{2} \right )$
$\left ( \alpha ,\beta \right )= \left ( 4,-5 \right )$