skills

You need to be familiar with symmetric combinations of polynomial roots

Question

Let $P(x)$ and $Q(x)$, as defined below, be two quadratic equations such that they share a common root, $\lambda$. Determine an expression for $\lambda$ in terms of the coefficients of $P(x)$ and $Q(x)$ $$ P(x) = x^2 + P_1x + P_0,\; Q(x) = x^2 + Q_1x + Q_0 $$

answer --- press button to toggle display

$\lambda = -\frac{P_0-Q_0}{P_1- Q_1}$

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$$P(x) =x^2 + P_1x + P_0 = x^2 - \left(\lambda+\mu\right)x + \lambda\mu$$ $$Q(x) = x^2 + Q_1x + Q_0 = x^2 - \left(\lambda + \nu\right)x+ \lambda\nu$$ $$ \lambda = \frac{\lambda\mu - \lambda\nu}{\mu - \nu} = -\frac{P_0-Q_0}{P_1- Q_1} $$