Solve for x: x³ - 5 x² - 2 x + 24 = 0. given that product of two roots is 12.?

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  • 7 months ago

    Let the roots be  α,  β,  γ. then --

     α + β + γ  =  5,   α β +  β γ +  α γ  =  - 2,   α β γ  = - 24  and  α β = 12 (given)

    hence  γ = - 2, 

    =>  α + β - 2 = 5 ........... =>   α + β  =  7

    We know that  ( α - β )²  =  ( α + β )² - 4 α β

    =>  ( α - β )² = ( 7 )² - 4 (12)  =  1

    =>  ( α - β )  =  ± 1

    Hence    α = 4 , β = 3,  Or   α = 3 , β = 4

    Hence the roots  are  4, 3, -2  ............. Answer  

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