Let p and q are the roots of x^2 − 2mx − 5n = 0 and m and n are the roots of x^2 − 2px − 5q = 0 If p ≠ q ≠ m ≠ n, then the value of p + q + m + n is ...


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Question Number 31320 by Joel578 last updated on 06/Mar/18

$Let p and q are the roots of$ $x^{2} - 2 m x - 5 n = 0$ $and m and n are the roots of$ $x^{2} - 2 p x - 5 q = 0$ $If p \neq q \neq m \neq n, then the value of$ $p + q + m + n is . . .$
	Answered by MJS last updated on 06/Mar/18

	$(x - p) (x - q) = x^{2} - 2 m x - 5 n$ $x^{2} - (p + q) x + p q = x^{2} - 2 m x - 5 n$ $p + q = 2 m; p q = - 5 n$ $m = \frac{p + q}{2}; n = - \frac{p q}{5}$ $(x - m) (x - n) = x^{2} - 2 p x - 5 q$ $x^{2} - (m + n) x + m n = x^{2} - 2 p x - 5 q$ $m + n = 2 p; m n = - 5 q$ $\frac{p + q}{2} - \frac{p q}{5} = 2 p \Rightarrow p = \frac{5 q}{15 + 2 q}$ $\frac{p + q}{2} \times \frac{p q}{5} = 5 q \Rightarrow q (p^{2} + p q - 50) = 0 \Rightarrow$ $\Rightarrow q_{1} = 0; p_{1} = 0 : not valid (p \neq q)$ $p^{2} + p q - 50 = 0 with p = \frac{5 q}{15 + 2 q}$ $q^{3} - 10 q^{2} - 300 q - 1225 = 0$ $try all \pm q with q ∣ 1225$ $\Rightarrow q_{2} = - 5; p_{2} = - 5; not valid (p \neq q)$ $q^{2} - 15 q - 225 = 0$ $q_{3} = \frac{15}{2} (1 - \sqrt{5}); p_{3} = \frac{5}{2} (3 + \sqrt{5})$ $m_{3} = \frac{5}{2} (3 - \sqrt{5}); n_{3} = \frac{15}{2} (1 + \sqrt{5})$ $q_{4} = n_{3}; p_{4} = m_{3}$ $m + n + p + q = 30$
	Commented by Joel578 last updated on 06/Mar/18

	$t h a n k y o u v e r y m u c h$
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