First slide

Theory of equations

Question

If the equations ${ax}^{2} + bx + c = 0 and x^{3} + 3 x^{2} + 3 x + 2 = 0$ have two common roots, then

Moderate

A

$a = b = c$
B

$a = b \neq c$
C

$a = - b = c$
D

none of these

Solution

By observation x = 1 satisfies equation $x^{3} + 3 x^{2} + 3 x + 2 = 0$
then we have $(x + 2) (x^{2} + x + 1) = 0$
$x^{2} + x + 1 = 0$ has nonreal roots.
Since nonreal roots occur in conjugate pair, $x^{2} + x + 1 = 0$
and ${ax}^{2} + bx + c = 0$ are identical
$\Rightarrow a = b = c$

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