First slide

Theory of equations

Question

$If the roots α, β of the equation p x^{2} + q x + r = 0 are real and of opposite signs (where p, q, r are real coefficients),$
$then the roots of the equation α (x - β)^{2} + β (x - α)^{2} = 0 a r e$

Moderate

A

positive
B

negative
C

real and of opposite signs
D

imaginary

Solution

$The equation α (x - β)^{2} + β (x - α)^{2} = 0$
$\begin{array}{ll} or (α + β) x^{2} - 2 x (α β + α β) + α β (α + β) = 0 \\ or (α + β) x^{2} - 4 α β x + α β (α + β) = 0 \end{array}$
$\begin{aligned} Product of its roots = \frac{α β (α + β)}{(α + β)} = α β < 0 g i v e n \\ Hence roots are real and of opposite signs. \end{aligned}$

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