First slide

Theory of equations

Question

$If 3 p^{2} = 5 p + 2 and 3 q^{2} = 5 q + 2 then the$
$equation whose roots 3 p - 2 q and 3 q - 2 p is$

Moderate

A

$x^{2} - 5 x + 100 = 0$
B

$3 x^{2} - 5 x - 100 = 0$
C

$3 x^{2} + 5 x + 100 = 0$
D

$5 x^{2} - x + 7 = 0$

Solution

: Note that p, q are roots of
$\begin{array}{l} 3 x^{2} = 5 x + 2 or 3 x^{2} - 5 x - 2 = 0 \\ p + q = 5 / 3, p q = - 2 / 3 . \end{array}$
Let $α = 3 p - 2 q, β = 3 q - 2 p,$
then $α = 3 p - 2 q, β = 3 q - 2 p,$
and $\begin{array}{l} α β = (3 p - 2 q) (3 q - 2 p) \\ = - 6 (p^{2} + q^{2}) + 13 p q \\ = - 6 (p + q)^{2} + 25 p q = - 100 / 3 \end{array}$
Thus, required quadratic equation is
$\begin{array}{l} x^{2} - (5 / 3) x - 100 / 3 = 0 \\ \Rightarrow 3 x^{2} - 5 x - 100 = 0 \end{array}$

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