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Give-the-function-x-2-7x-8-0-have-two-roots-x-1-and-x-2-No-solving-the-function-Find-x-1-3-x-2-2023-

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Question Number 192390 by TUN last updated on 16/May/23
Give the function: x^2 −7x−8=0 have two roots x_(1 ) and x_2 No solving the function Find: x_1 ^3 +x_2 +2023
Givethefunction:x2−7x−8=0havetworootsx1andx2NosolvingthefunctionFind:x13+x2+2023
Answered by AST last updated on 17/May/23
x_1 ^3 +x_2 +2023=p; x_2 ^3 +x_1 +2023=q p+q=(x_1 +x_2 )^3 −3x_1 x_2 (x_1 +x_2 )+x_1 +x_2 +4046 x_1 +x_2 =7;x_1 x_2 =−8⇒p+q=343+168+7+4046=4564 pq=(x_1 x_2 )^3 +x_1 ^4 +2023(x_1 ^3 +x_2 ^3 )+x_2 ^4 +x_1 x_2 +2023(x_2 +x_1 ) +2023^2 =−512+[(7^2 +16)^2 −128]+2023(511)−8+2023(7) +2023^2 =5144020 p+q=4464;pq=5144020⇒p(4464−p)=5144020 ⇒p=2030,q=2534 (up to symmetry)
x13+x2+2023=p;x23+x1+2023=qp+q=(x1+x2)3−3x1x2(x1+x2)+x1+x2+4046x1+x2=7;x1x2=−8⇒p+q=343+168+7+4046=4564pq=(x1x2)3+x14+2023(x13+x23)+x24+x1x2+2023(x2+x1)+20232=−512+[(72+16)2−128]+2023(511)−8+2023(7)+20232=5144020p+q=4464;pq=5144020⇒p(4464−p)=5144020⇒p=2030,q=2534(uptosymmetry)

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