First slide

Theory of expressions

Question

$If a, b, and c are not all equal and α and β$
$be the roots of the equation a x^{2} + b x + c = 0, then value of$
$(1 + α + α^{2}) (1 + β + β^{2}) is$

Moderate

A

0
B

positive
C

negative
D

non-negative

Solution

$We have α + β = - b / a, α β = c / a$
$\begin{array}{l} Now, (1 + α + α^{2}) (1 + β + β^{2}) \\ = 1 + α + β + α^{2} + β^{2} + α β + α^{2} β + α β^{2} + α^{2} β^{2} \\ = 1 + (α + β) + (α + β)^{2} - 2 α β + α β [1 + α + β + α β] \\ = 1 - \frac{b}{a} + \frac{b^{2}}{a^{2}} - \frac{2 c}{a} + \frac{c}{a} + \frac{c}{a} (- \frac{b}{a}) + \frac{c^{2}}{a^{2}} \\ = \frac{1}{a^{2}} \{a^{2} + b^{2} + c^{2} - b c - c a - a b\} \\ = \frac{1}{2 a^{2}} [(b - c)^{2} + (c - a)^{2} + (a - b)^{2}] \geq 0 \end{array}$

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