If the sum of zeroes of a quadratic polynomial $f (t) = k t^{2} + 2 t + 3 k$ is equal to their product, find the value of k.

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Solution

Given quadratic polynomial,
$f (t) = k t^{2} + 2 t + 3 k$
$α, β$ are the roots of the quadratic polynomial, then
$α + β = - \frac{2}{k}$ and $α β = \frac{3 k}{k}$
According to the condition $α + β = α β$
$⟹ - \frac{2}{k} = 3 ⟹ k = - \frac{2}{3}$

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