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Question Number 166419 by ajfour last updated on 19/Feb/22

Commented by ajfour last updated on 19/Feb/22

$B l u e c u r v e y = x^{3} - x$ $R e d c u r v e y = (x - h) - {(x - h)}^{3} + k$ $F i n d x = p w h e n y = c .$
	Answered by ajfour last updated on 20/Feb/22

	$y = (x - h) - {(x - h)}^{3} + k$ $y = x^{3} - x$ $s u b t r a c t i n g f o r i n t e r s e c t i o n p o i n t s$ $2 x - h - x^{3} - {(x - h)}^{3} + k = 0$ $\Rightarrow 2 x^{3} - 3 {h x}^{2} + (3 h^{2} - 2) x - h^{3} + h + k = 0$ $r o o t s a r e p, s, s$ $\Rightarrow 2 {(x - s)}^{2} (x - p) = 0$ $2 x^{3} - 2 (p + 2 s) x^{2} + 2 s (2 p + s) x$ $- 2 {p s}^{2} = 0$ $\Rightarrow p + 2 s = \frac{3 h}{2}$ $2 s (2 p + s) = 3 h^{2} - 2$ $2 {p s}^{2} = h^{3} - h - k$ $\Rightarrow 2 s (2 p + s) = 3 {(\frac{2 p}{3} + \frac{4 s}{3})}^{2} - 2$ $\Rightarrow 12 s p + 6 s^{2} + 6$ $= 4 p^{2} + 16 s^{2} + 16 s p$ $\Rightarrow 4 p^{2} + 4 s p + 10 s^{2} - 6 = 0$ $\Rightarrow 4 (p + c) + 4 {s p}^{2} + 2 p (5 s^{2} - 3) = 0$ $\Rightarrow$ $4 p + 4 c + 2 p (5 s^{2} - 3)$ $= 4 s^{2} p + 2 s (5 s^{2} - 3)$ $\Rightarrow p = \frac{2 s (5 s^{2} - 3) - 4 c}{2 (5 s^{2} - 3) - 4 s^{2} + 4}$ $o r p = \frac{s (5 s^{2} - 3) - 2 c}{3 s^{2} - 1}$ $4 {(5 s^{3} - 3 s - 2 c)}^{2} + 4 s (5 s^{3} - 3 s - 2 c) (3 s^{2} - 1)$ $+ 2 (5 s^{2} - 3) {(3 s^{2} - 1)}^{2} = 0$ $\Rightarrow 250 s^{6} - 290 s^{4} - 104 {c s}^{3}$ $+ 94 x^{2} + 56 c x + 16 c^{2} - 6 = 0$ $c a n n o t w e f a c t o r i s e t h i s e q . . ?$
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