x^4 +ax^3 +bx^2 +cx+d=0 ⇒ x^2 +(d/x^2 )+ax+(c/x)+b=0 say (1/x)=t ⇒ x^2 +ax+b+dt^2 +ct=0 let x^2 +ax+p=0 & dt^2 +ct+q=0 p+q=b x=−(a/2)±(√((a^2 /4)−p)) t=−(c/(2d))±(√((c^2 /(4d^2 ))−(q/d))) tx=1 ⇒ ((ac)/(4d))∓(a/2)(√((c^2 /(4d^2 ))−(q/d)))∓(c/(2d))(√((a^2 /4)−p)) +(√((a^2 /4)−p))(√((c^2 /(4d^2 ))−(q/d))) =1 let p=(a^2 /4) ⇒ ((ac)/(4d))∓(a/2)(√((c^2 /(4d^2 ))−(b/d)+(a^2 /(4d))))=1 ⇒ (c^2 /(4d^2 ))−(b/d)+(a^2 /(4d))=(4/a^2 )+(c^2 /(4d^2 ))−((2c)/(ad)) ⇒ 4a^2 b−a^4 +16d−8ac=0 if x=s+h s^4 +4hs(s^2 +h^2 )+6h^2 s^2 +h^4 +a{s^3 +3hs(s+h)+h^2 } +b{s^2 +2hs+h^2 }+c(s+h)+d=0 A=4h+a B=6h^2 +3ah+b C=4h^3 +3ah^2 +2bh+c D=h^4 +ah^3 +bh^2 +ch+d ⇒ (4h+a)^2 (8h^2 +4ah+4b−a^2 ) +16(h^4 +ah^3 +bh^2 +ch+d) =8(4h+a)(4h^3 +3ah^2 +2bh+c) .....


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Question Number 164019 by ajfour last updated on 13/Jan/22
$x^4 +ax^3 +bx^2 +cx+d=0 ⇒ x^2 +(d/x^2 )+ax+(c/x)+b=0 say (1/x)=t ⇒ x^2 +ax+b+dt^2 +ct=0 let x^2 +ax+p=0 & dt^2 +ct+q=0 p+q=b x=−(a/2)±(√((a^2 /4)−p)) t=−(c/(2d))±(√((c^2 /(4d^2 ))−(q/d))) tx=1 ⇒ ((ac)/(4d))∓(a/2)(√((c^2 /(4d^2 ))−(q/d)))∓(c/(2d))(√((a^2 /4)−p)) +(√((a^2 /4)−p))(√((c^2 /(4d^2 ))−(q/d))) =1 let p=(a^2 /4) ⇒ ((ac)/(4d))∓(a/2)(√((c^2 /(4d^2 ))−(b/d)+(a^2 /(4d))))=1 ⇒ (c^2 /(4d^2 ))−(b/d)+(a^2 /(4d))=(4/a^2 )+(c^2 /(4d^2 ))−((2c)/(ad)) ⇒ 4a^2 b−a^4 +16d−8ac=0 if x=s+h s^4 +4hs(s^2 +h^2 )+6h^2 s^2 +h^4 +a{s^3 +3hs(s+h)+h^2 } +b{s^2 +2hs+h^2 }+c(s+h)+d=0 A=4h+a B=6h^2 +3ah+b C=4h^3 +3ah^2 +2bh+c D=h^4 +ah^3 +bh^2 +ch+d ⇒ (4h+a)^2 (8h^2 +4ah+4b−a^2 ) +16(h^4 +ah^3 +bh^2 +ch+d) =8(4h+a)(4h^3 +3ah^2 +2bh+c) .....$
$x^{4} + {a x}^{3} + {b x}^{2} + c x + d = 0$ $\Rightarrow x^{2} + \frac{d}{x^{2}} + a x + \frac{c}{x} + b = 0$ $s a y \frac{1}{x} = t \Rightarrow$ $x^{2} + a x + b + {d t}^{2} + c t = 0$ $l e t x^{2} + a x + p = 0$ $& {d t}^{2} + c t + q = 0$ $p + q = b$ $x = - \frac{a}{2} \pm \sqrt{\frac{a^{2}}{4} - p}$ $t = - \frac{c}{2 d} \pm \sqrt{\frac{c^{2}}{4 d^{2}} - \frac{q}{d}}$ $t x = 1 \Rightarrow$ $\frac{a c}{4 d} \mp \frac{a}{2} \sqrt{\frac{c^{2}}{4 d^{2}} - \frac{q}{d}} \mp \frac{c}{2 d} \sqrt{\frac{a^{2}}{4} - p}$ $+ \sqrt{\frac{a^{2}}{4} - p} \sqrt{\frac{c^{2}}{4 d^{2}} - \frac{q}{d}} = 1$ $l e t p = \frac{a^{2}}{4} \Rightarrow$ $\frac{a c}{4 d} \mp \frac{a}{2} \sqrt{\frac{c^{2}}{4 d^{2}} - \frac{b}{d} + \frac{a^{2}}{4 d}} = 1$ $\Rightarrow \frac{c^{2}}{4 d^{2}} - \frac{b}{d} + \frac{a^{2}}{4 d} = \frac{4}{a^{2}} + \frac{c^{2}}{4 d^{2}} - \frac{2 c}{a d}$ $\Rightarrow 4 a^{2} b - a^{4} + 16 d - 8 a c = 0$ $i f x = s + h$ $s^{4} + 4 h s (s^{2} + h^{2}) + 6 h^{2} s^{2} + h^{4}$ $+ a {s^{3} + 3 h s (s + h) + h^{2}}$ $+ b {s^{2} + 2 h s + h^{2}} + c (s + h) + d = 0$ $A = 4 h + a$ $B = 6 h^{2} + 3 a h + b$ $C = 4 h^{3} + 3 {a h}^{2} + 2 b h + c$ $D = h^{4} + {a h}^{3} + {b h}^{2} + c h + d$ $\Rightarrow$ ${(4 h + a)}^{2} (8 h^{2} + 4 a h + 4 b - a^{2})$ $+ 16 (h^{4} + {a h}^{3} + {b h}^{2} + c h + d)$ $= 8 (4 h + a) (4 h^{3} + 3 {a h}^{2} + 2 b h + c)$ $. . . . .$
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