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x-2-2000x-1-0-Roots-a-and-b-x-2-2008x-1-0-Roots-c-and-d-Find-a-c-b-d-a-d-b-c-




Question Number 215072 by hardmath last updated on 27/Dec/24
x^2  + 2000x + 1 = 0  Roots:  a  and  b  x^2  − 2008x − 1 = 0  Roots:  c  and  d  Find:  (a+c)(b+d)(a−d)(b−c) = ?
x2+2000x+1=0Roots:aandbx22008x1=0Roots:canddFind:(a+c)(b+d)(ad)(bc)=?
Commented by TonyCWX08 last updated on 28/Dec/24
Next time.  Have your answer as ANSWER.  Not COMMENT.
Nexttime.HaveyouranswerasANSWER.NotCOMMENT.
Commented by Abdullahrussell last updated on 28/Dec/24
 (a+c)(b+d)(a−d)(b−c)   =(ab+ad+bc+cd)(ab−ac−bd+cd)  =(1+ad+bc−1)(1−ac−bd−1)  =−(ad+bc)(ac+bd)  =−(a^2 cd+abd^2 +abc^2 +b^2 cd)  =−(−a^2 +d^2 +c^2 −b^2 )  =a^2 +b^2 −(c^2 +d^2 )  =(a+b)^2 −2ab−(c+d)^2 +2cd  =(−2000)^2 −2−(2008)^2 −2  =(2000+2008)(2000−2008)−4  =4008×−8−4  =−32064−4=−32068
(a+c)(b+d)(ad)(bc)=(ab+ad+bc+cd)(abacbd+cd)=(1+ad+bc1)(1acbd1)=(ad+bc)(ac+bd)=(a2cd+abd2+abc2+b2cd)=(a2+d2+c2b2)=a2+b2(c2+d2)=(a+b)22ab(c+d)2+2cd=(2000)22(2008)22=(2000+2008)(20002008)4=4008×84=320644=32068
Commented by MathematicalUser2357 last updated on 28/Dec/24
You better not to be angry
Commented by mr W last updated on 29/Dec/24
i think he is not angry. he is just   giving an useful suggestion.
ithinkheisnotangry.heisjustgivinganusefulsuggestion.
Answered by TonyCWX08 last updated on 28/Dec/24
a+b=−2000  ab=1  c+d=2008  cd=−1    (a+c)(b+d)  =ab+ad+bc+cd  =ad+bc    (a−d)(b−c)  =ab−ac−bd+cd  =−ac−bd    (ad+bc)(−ac−bd)  =−a^2 cd−abd^2 −ac^2 b−b^2 cd  =a^2 −d^2 −c^2 +b^2   =a^2 +b^2 −(c^2 +d^2 )  =((a+b)^2 −2ac)−((c+d)^2 −2cd)  =2000^2 −2−(2008^2 +2)  =2000^2 −2008^2 −4  =−32068
a+b=2000ab=1c+d=2008cd=1(a+c)(b+d)=ab+ad+bc+cd=ad+bc(ad)(bc)=abacbd+cd=acbd(ad+bc)(acbd)=a2cdabd2ac2bb2cd=a2d2c2+b2=a2+b2(c2+d2)=((a+b)22ac)((c+d)22cd)=200022(20082+2)=20002200824=32068

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