Question

If α,β are the zeroes of the quadratic polynomial ax2+bx+c then find the value of α2+β2?

Answer 1

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Answer 2

Answer:

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

Step-by-step explanation:

If $\alpha$ and $\beta$ are the roots of the  $ax^{2} +bx +c = 0$ , then we have,

Sum of roots = $\alpha +\beta = \frac{-b}{a}$  and

Product of roots = $\alpha \beta = \frac{c}{a}$

Required to find $\alpha ^{2} + \beta ^{2}$

We know that,  $(a+b)^{2} = a^{2} + b^{2} +2ab,$

$a^{2} + b^{2} = (a+b)^{2} -2ab$ ---------------(1)

Substituting a = $\alpha$ and b = $\beta$ in the above equation

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

Substituting the value of $\alpha +\beta$ and $\alpha \beta$ in the above equation we get,

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

= $\frac{b^2}{a^2} - \frac{2c}{a}$

= $\frac{b^2 - 2ac}{a^2}$

Hence,   $\alpha ^2 + \beta ^2 = \frac{b^2 - 2ac}{a^2}$