First slide

Theory of equations

Question

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, then it must be equal to

Moderate

A

$\frac{pq' - p' q}{q - q'}$
B

$\frac{q - q'}{p' - p}$
C

$\frac{p' - p}{q - q'}$
D

$\frac{pq' - p' q}{p - p'}$

Solution

Equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root. Therefore,
${(q - q')}^{2} = (pq' - p' q) (p' - p)$ (1)
Subtracting two equations, we have
$x = \frac{q - q'}{p' - p}$
Also using (1),
$x = \frac{q - q'}{p' - p} = \frac{pq' - p' q}{q - q'}$

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