Question

If α and β are the zeroes of the polynomial ax2 + bx + c, find the value of α2 + β2.​

Answer 1

Given :-

α and β are the roots of Quadratic polynomial ax² + bx + c

To find :-

$\alpha {}^{2} + \beta {}^{2}$

Solution :-

$ax {}^{2} + bx + c$

As we know relation between sum of zeroes and product of zeroes

Sum of zeroes :-

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

Product of zeroes

$\alpha \beta = \dfrac{c}{a}$

We have to find the value of ${\alpha^2 + \beta^2}$

As we know (a+b)² = a² + b² + 2ab

But we have to find value of a² + b²

a² + b² = (a+b)² -2ab

Similarly,

$\alpha {}^{2} + \beta { }^{2} = ( \alpha + \beta ) {}^{2} - 2 \alpha \beta$

Plugging the values

${\;\;\alpha^{2}\;+\;\beta^{2}\;=\;\bigg(\dfrac{-b}{a}\bigg)^{2}\;-\;2\bigg(\dfrac{c}{a}\bigg)}$

$\alpha {}^{2} + \beta {}^{2} = \dfrac{b {}^{2} }{a {}^{2} } - \dfrac{2c}{a}$

$\alpha {}^{2} + \beta {}^{2} = \dfrac{b { }^{2} - 2ac}{a {}^{2} }$

So, the value of α² + β² is b²-2ac/a²