x^3 +ax^2 +bx+c=0 Let x=((pt+q)/(t+1)) p^3 t^3 +3p^2 qt^2 +3pq^2 t+q^3 +a(t+1)(p^2 t^2 +2pqt+q^2 ) +b(pt+q)(t^2 +2t+1) +c(t^3 +3t^2 +3t+1) = 0 ⇒ (p^3 +ap^2 +bp+c)t^3 +(3p^2 q+ap^2 +2apq+bq+2bp+3c)t^2 +(3q^2 p+aq^2 +2apq+bp+2bq+3c)t +(q^3 +aq^2 +bq+c) = 0 Let coeffs. of t^2 and t be zero. Subtracting and adding them 3pq+a(p+q)+b=0 & 3pq(p+q)+a{(p+q)^2 −2pq} +4apq+3b(p+q)+6c = 0 lets call pq=m , p+q=s ⇒ 3m+as+b=0 ....(i) 3ms+a(s^2 −2m)+4am+3bs+6c=0 ⇒ am+bs+3c=0 ....(ii) ⇒ s=((9c−ab)/(a^2 −3b)) ; m=((b^2 −3ac)/(a^2 −3b)) Now p,q are roots of eq. z^2 −sz+m=0 p,q = (s/2)±(√((s^2 /4)−m)) t^3 =−(((q^3 +aq^2 +bq+c)/(p^3 +ap^2 +bq+c))) (t≠−1) x=((pt+q)/(t+1)) .


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Question Number 74339 by ajfour last updated on 22/Nov/19
$x^3 +ax^2 +bx+c=0 Let x=((pt+q)/(t+1)) p^3 t^3 +3p^2 qt^2 +3pq^2 t+q^3 +a(t+1)(p^2 t^2 +2pqt+q^2 ) +b(pt+q)(t^2 +2t+1) +c(t^3 +3t^2 +3t+1) = 0 ⇒ (p^3 +ap^2 +bp+c)t^3 +(3p^2 q+ap^2 +2apq+bq+2bp+3c)t^2 +(3q^2 p+aq^2 +2apq+bp+2bq+3c)t +(q^3 +aq^2 +bq+c) = 0 Let coeffs. of t^2 and t be zero. Subtracting and adding them 3pq+a(p+q)+b=0 & 3pq(p+q)+a{(p+q)^2 −2pq} +4apq+3b(p+q)+6c = 0 lets call pq=m , p+q=s ⇒ 3m+as+b=0 ....(i) 3ms+a(s^2 −2m)+4am+3bs+6c=0 ⇒ am+bs+3c=0 ....(ii) ⇒ s=((9c−ab)/(a^2 −3b)) ; m=((b^2 −3ac)/(a^2 −3b)) Now p,q are roots of eq. z^2 −sz+m=0 p,q = (s/2)±(√((s^2 /4)−m)) t^3 =−(((q^3 +aq^2 +bq+c)/(p^3 +ap^2 +bq+c))) (t≠−1) x=((pt+q)/(t+1)) .$
$x^{3} + {a x}^{2} + b x + c = 0$ $L e t x = \frac{p t + q}{t + 1}$ $p^{3} t^{3} + 3 p^{2} {q t}^{2} + 3 {p q}^{2} t + q^{3}$ $+ a (t + 1) (p^{2} t^{2} + 2 p q t + q^{2})$ $+ b (p t + q) (t^{2} + 2 t + 1)$ $+ c (t^{3} + 3 t^{2} + 3 t + 1) = 0$ $\Rightarrow$ $(p^{3} + {a p}^{2} + b p + c) t^{3}$ $+ (3 p^{2} q + {a p}^{2} + 2 a p q + b q + 2 b p + 3 c) t^{2}$ $+ (3 q^{2} p + {a q}^{2} + 2 a p q + b p + 2 b q + 3 c) t$ $+ (q^{3} + {a q}^{2} + b q + c) = 0$ $L e t c o e f f s . o f t^{2} a n d t b e z e r o .$ $S u b t r a c t i n g a n d a d d i n g t h e m$ $3 p q + a (p + q) + b = 0 &$ $3 p q (p + q) + a {{(p + q)}^{2} - 2 p q}$ $+ 4 a p q + 3 b (p + q) + 6 c = 0$ $l e t s c a l l p q = m, p + q = s \Rightarrow$ $3 m + a s + b = 0 . . . . (i)$ $3 m s + a (s^{2} - 2 m) + 4 a m + 3 b s + 6 c = 0$ $\Rightarrow a m + b s + 3 c = 0 . . . . (i i)$ $\Rightarrow s = \frac{9 c - a b}{a^{2} - 3 b}; m = \frac{b^{2} - 3 a c}{a^{2} - 3 b}$ $N o w p, q a r e r o o t s o f e q .$ $z^{2} - s z + m = 0$ $p, q = \frac{s}{2} \pm \sqrt{\frac{s^{2}}{4} - m}$ $t^{3} = - (\frac{q^{3} + {a q}^{2} + b q + c}{p^{3} + {a p}^{2} + b q + c}) (t \neq - 1)$ $x = \frac{p t + q}{t + 1} .$
Commented by ajfour last updated on 23/Nov/19
$considering x^3 −6x^2 +3x+10=0 a=−6, b=3, c=10 s=((90+18)/(27))=4 , m=((9+180)/(27))=7 p,q=2±i(√3) ; let p=2−i(√3) let t^3 =−(((8+12i(√3)−18−3(√3)i−6(1+4i(√3))+3(2+i(√3))+10)/(8−12i(√3)−18+3(√3)i−6(1−4i(√3))+3(2−i(√3))+10))) ⇒ t^3 = −(((−12(√3)i)/(12(√3)i))) ⇒ t=1,ω,ω^2 x_1 =(4/2) = 2 x_2 =(((2−i(√3))(−1+i(√3))/2+(2+i(√3)))/((1+i(√3))/2)) = (((1+3(√3)i)+(4+2(√3)i))/(1+i(√3))) = 5 x_3 =(((2−i(√3))(−1−i(√3))/2+(2+i(√3)))/((1−i(√3))/2)) =((−1+i(√3))/( 1−i(√3))) = −1 hence roots are x∈{2,5,−1} (true indeed!)$
$c o n s i d e r i n g x^{3} - 6 x^{2} + 3 x + 10 = 0$ $a = - 6, b = 3, c = 10$ $s = \frac{90 + 18}{27} = 4, m = \frac{9 + 180}{27} = 7$ $p, q = 2 \pm i \sqrt{3}; l e t p = 2 - i \sqrt{3}$ $l e t t^{3} = - (\frac{8 + 12 i \sqrt{3} - 18 - 3 \sqrt{3} i - 6 (1 + 4 i \sqrt{3}) + 3 (2 + i \sqrt{3}) + 10}{8 - 12 i \sqrt{3} - 18 + 3 \sqrt{3} i - 6 (1 - 4 i \sqrt{3}) + 3 (2 - i \sqrt{3}) + 10})$ $\Rightarrow t^{3} = - (\frac{- 12 \sqrt{3} i}{12 \sqrt{3} i}) \Rightarrow t = 1, ω, ω^{2}$ $x_{1} = \frac{4}{2} = 2$ $x_{2} = \frac{(2 - i \sqrt{3}) (- 1 + i \sqrt{3}) / 2 + (2 + i \sqrt{3})}{(1 + i \sqrt{3}) / 2}$ $= \frac{(1 + 3 \sqrt{3} i) + (4 + 2 \sqrt{3} i)}{1 + i \sqrt{3}} = 5$ $x_{3} = \frac{(2 - i \sqrt{3}) (- 1 - i \sqrt{3}) / 2 + (2 + i \sqrt{3})}{(1 - i \sqrt{3}) / 2}$ $= \frac{- 1 + i \sqrt{3}}{1 - i \sqrt{3}} = - 1$ $h e n c e r o o t s a r e x \in {2, 5, - 1}$ $(t r u e i n d e e d!)$
Commented by ajfour last updated on 23/Nov/19

$c o n s i d e r i n g x^{3} - 6 x^{2} + 3 x + 10 = 0$ $a g a i n a n d u s i n g {C a r d a n o}^{'} s m e t h o d$ $l e t x = t + 2 \Rightarrow$ $t^{3} + 12 t + 8 - 24 t - 24 + 3 t + 6 + 10 = 0$ $\Rightarrow t^{3} - 9 t = 0$ $\Rightarrow t = \pm 3, 0$ $\Rightarrow x = - 1, 2, 5 (e a s y e n o u g h,$ $n q u i t e d i s h e a r t n i n g!)$
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