First slide

Theory of expressions

Question

The real numbers $α, β$ are of opposite signs and $| α | \neq | β | .$ The roots of $α {(x - β)}^{2} + β {(x - α)}^{2}$ $= 0$ are

Moderate

A

Positive
B

Negative
C

Real and of opposite signs
D

Non-real

Solution

The given equation is $α {(x - β)}^{2} + β {(x - α)}^{2}$ $= 0$
$\Rightarrow α (x^{2} + β^{2} - 2 x α β) + β (x^{2} + α^{2} - 2 α x)$

$\Rightarrow x^{2} (α + β) - 4 α β x + α β (α + β) = 0$

Or $x^{2} - (\frac{4 α β}{α + β}) x + α β = 0$

Here, coefficient of $x^{2}$ and the constant term are of opposite signs $(∵ α β < 0)$

Therefore, the roots of the given equation are real and of opposite signs.

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