First slide

Theory of equations

Question

If a, b, c are positive real numbers such that the equations $a x^{2} + b x + c = 0 and b x^{2} + c x + a = 0$ have a common root then

Moderate

A

$a + b w + c w^{2} = 0$
B

$a + b w^{2} + c w = 0$
C

$a^{3} + b^{3} + c^{3} = 3 a b c$
D

all the above

Solution

Let a be the common root of the two equations
Then
and $\begin{aligned} a α^{2} + b α + c = 0 \\ b α^{2} + c α + a = 0 \end{aligned}$
Solving these two equations, we get
$\begin{array}{l} α^{2} = \frac{α}{a b - c^{2}} \\ \Rightarrow a^{2} = \frac{a b - a^{2}}{a c - b^{2}} and α = \frac{1}{a c - b^{2}} \\ \Rightarrow \frac{a b - c^{2}}{a c - b^{2}} = {(\frac{b c - a^{2}}{a c - b^{2}})}^{2} \\ \Rightarrow (a b - c^{2}) (a c - b^{2}) = {(b c - a^{2})}^{2} \\ \Rightarrow a (a^{3} + b^{3} + c^{3} - 3 a b c) = 0 \\ \Rightarrow a^{3} + b^{3} + c^{3} - 3 a b c = 0 \\ \Rightarrow (a + b + c) (a + b w + c w^{2}) \\ \Rightarrow a + b w + c w^{2} = 0 or a + b w^{2} + c w = 0 [∵ a + b + c \neq 0] \end{array}$

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