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Question Number 192855 by pascal889 last updated on 29/May/23

	Answered by a.lgnaoui last updated on 29/May/23
	${ ((P^2 −2aP−10P+2a^2 +6a −6=0 (1))),((P^2 −27P +27a−27=0 (2) )) :} (1)−(2)⇒ 2a^2 −21a+21+P(17−2a)=0 ⇒ P=((2a^2 −21a+21)/(2a−17)) (3) ⇒(2)((2(a^2 −((21)/2)a+((21)/2)))/(2a−17))=(((a−((21)/4))^2 −(((63)/(16))))/(a−((17)/2))) P=(((a−((21)/4))^2 −((63)/(16)))/(a−((17)/2))) P^2 =(((a−((21)/4))^4 +((63)/(16^2 ))−((63)/8)(a−((21)/4)))/((a−((17)/2))^2 )) (2) P^2 =27(P−a+1) =27((((a−((21)/4))^2 −((63)/(16)))/(a−((17)/2)))−a+1) =27[(((4a−21)^2 −63)/(8(2a−17)))−(((16a−136)(a+1))/(8(2a−17)))] =27[((21^2 −168a−16a +136a +136−63)/(8(2a−17))] P^2 =27×((38−6a)/(2a−17)) (l)⇔P^2 −2(a+5)P+2(a^2 +3a−3)=0 ((27(38−6a))/(2a−17))−2(a+5)(((4a−21)^2 −63)/(4(2a−17)))+2(a^2 +3a−3)=0 4×27(38−6a)−2(a+5)[4a−21)^2 −63] +8(a^2 +3a−3)=0 998−162a−(2a+10)(16a^2 −168a−163) +8(a^2 +3a−3)(2a−17)=0 (998+1630−408)−16a^3 − (336−160−136−48)a^2 (+326−162 +1680−408−48)a 4a^3 +2a^2 −347a+555=0 ⇒a={−10,226633 ; 1,669079 ;8,097282} ⇒P=f(a)=((2a^2 −21a+21)/(2a−17))$
	${\begin{cases} P^{2} - 2 aP - 10 P + {2 a}^{2} + 6 a - 6 = 0 (1) \\ P^{2} - 27 P + 27 a - 27 = 0 (2) \end{cases}$ $(1) - (2) \Rightarrow {2 a}^{2} - 21 a + 21 + P (17 - 2 a) = 0$ $\Rightarrow P = \frac{2 a^{2} - 21 a + 21}{2 a - 17} (3)$ $\Rightarrow (2) \frac{2 (a^{2} - \frac{21}{2} a + \frac{21}{2})}{2 a - 17} = \frac{{(a - \frac{21}{4})}^{2} - (\frac{63}{16})}{a - \frac{17}{2}}$ $P = \frac{{(a - \frac{21}{4})}^{2} - \frac{63}{16}}{a - \frac{17}{2}}$ $P^{2} = \frac{{(a - \frac{21}{4})}^{4} + \frac{63}{16^{2}} - \frac{63}{8} (a - \frac{21}{4})}{{(a - \frac{17}{2})}^{2}}$ $(2) P^{2} = 27 (P - a + 1)$ $= 27 (\frac{{(a - \frac{21}{4})}^{2} - \frac{63}{16}}{a - \frac{17}{2}} - a + 1)$ $= 27 [\frac{{(4 a - 21)}^{2} - 63}{8 (2 a - 17)} - \frac{(16 a - 136) (a + 1)}{8 (2 a - 17)}]$ $= 27 [\frac{21^{2} - 168 a - 16 a + 136 a + 136 - 63}{8 (2 a - 17}]$ $P^{2} = 27 \times \frac{38 - 6 a}{2 a - 17}$ $(l) \Leftrightarrow P^{2} - 2 (a + 5) P + 2 (a^{2} + 3 a - 3) = 0$ $\frac{27 (38 - 6 a)}{2 a - 17} - 2 (a + 5) \frac{{(4 a - 21)}^{2} - 63}{4 (2 a - 17)} + 2 (a^{2} + 3 a - 3) = 0$ $4 \times 27 (38 - 6 a) - 2 (a + 5) {[4 a - 21)}^{2} - 63]$ $+ 8 (a^{2} + 3 a - 3) = 0$ $998 - 162 a - (2 a + 10) ({16 a}^{2} - 168 a - 163)$ $+ 8 (a^{2} + 3 a - 3) (2 a - 17) = 0$ $(998 + 1630 - 408) - {16 a}^{3} -$ $(336 - 160 - 136 - 48) a^{2} (+ 326 - 162$ $+ 1680 - 408 - 48) a$ $4 a^{3} + 2 a^{2} - 347 a + 555 = 0$ $\Rightarrow a = {- 10, 226633; 1, 669079; 8, 097282}$ $\Rightarrow P = f (a) = \frac{2 a^{2} - 21 a + 21}{2 a - 17}$
	Answered by York12 last updated on 30/May/23

	$p^{2} + 2 a^{2} - 2 a p + 6 a - 10 p - 6 = 0 . . . . . . (i)$ $p^{2} - 27 p + 27 a - 27 = 0 . . . . . . . (i i)$ $f r o m (i i)$ $p^{2} - 27 (p - a + 1) \to (I)$ $t h e n (i) c a n b e w r i t t e n a s$ $p^{2} - 27 (p - a + 1) + 17 p - 21 a + 21 + 2 a^{2} - 2 a p$ $f r o m (I)$ $17 p - 21 a + 21 + 2 a^{2} - 2 a p = 0$ $2 a p - 17 p - 2 a^{2} + 21 a - 21 = 0$ $p (2 a - 17) = 2 a^{2} - 21 a + 21$ $p = \frac{2 a^{2} - 21 a + 21}{2 a - 17}$
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