First slide

Theory of equations

Question

If every pair from among the equations $x^{2} + ax + bc = 0, x^{2} + bx + ca = 0, and x^{2} + cx + ab = 0$ has a common root, then

Moderate

A

the sum of the three common roots is $- \frac{1}{2}$ (a + b + c)
B

the sum of the three common roots is 2(a + b + c)
C

one of the values of the product of the three common roots is abc
D

the product of the three common roots is a²b²C²

Solution

Since each pair has common root, let the roots be $α, β$ for
Eq. (1): $β, γ$ for Eq.(2) and $γ, α$ for Eq. (3). Therefore,
$α + β = - a, αβ = bc$
$\begin{array}{l} β + γ = - b, βγ = ca \\ γ + α = - c, γα = ab \end{array}$
Adding, we get
$\begin{aligned} 2 (α + β + γ) = - (a + b + c) \\ \Rightarrow α + β + γ = - \frac{1}{2} (a + b + c) \end{aligned}$
Also by multiplying product of roots, we have
$α^{2} β^{2} γ^{2} = a^{2} b^{2} c^{2} or αβγ = abc$

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