The number of triangles formed by the curves $x^{3} - x^{2} - x - 2 = 0$ and $4 x^{2} + x^{3} + 12 x + 8 = 0$ is equal to ?

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Solution

$4 x^{2} + x^{3} + 12 x + 8 = 0 \dots (1)$
$x^{3} = x^{2} + x + 2 \dots (2)$
Put $(2)$ in $(1)$
$⟹ 5 x^{2} + 13 x + 10 = 0 ⟹ x$ has only $2$ values
but triangle needs three points so, no triangle is formed

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