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Persamaan kuadrat x 2 + kx − ( 2 k + 4 ) = 0 mempunyai akar-akar α dan β. Jika α 2 + β 2 = 53 , nilai k yang memenuhi adalah ....

Persamaan kuadrat $x^{2} + kx - (2 k + 4) = 0$ mempunyai akar-akar α dan β. Jika $α^{2} + β^{2} = 53$ , nilai k yang memenuhi adalah ....

k = -15 atau k = 3
k = -9 atau k = -5
k = 9 atau k = 5
k = -9 atau k = 5
k = 9 atau k = -5

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$begin mathsize 14px style straight i right parenthesis space Cari space hubungan space akar minus akar space persamaan space kuadrat space colon space space space Untuk space persamaan space kuadrat space ax squared plus bx plus straight c equals 0 space yang space memiliki space space space space akar minus akar space straight alpha space dan space straight beta comma space maka space space space straight alpha plus straight beta equals negative straight b over straight a space space space αβ equals straight c over straight a space space space Persamaan space kuadrat space straight x squared plus kx minus left parenthesis 2 straight k plus 4 right parenthesis equals 0 space dengan space straight a equals 1 comma space straight b equals straight k comma space straight c equals negative left parenthesis 2 straight k plus 4 right parenthesis comma space space space memiliki space akar minus akar space straight alpha space dan space straight beta comma space maka space space space straight alpha plus straight beta equals negative straight b over straight a equals negative straight k over 1 equals negative straight k space space space αβ equals straight c over straight a equals fraction numerator negative open parentheses 2 straight k plus 4 close parentheses over denominator 1 end fraction equals negative open parentheses 2 straight k plus 4 close parentheses ii right parenthesis space Selesaikan space persamaan space yang space diberikan comma space yaitu space straight alpha squared plus straight beta squared equals 53. space Ingat space space space space straight alpha squared plus straight beta squared equals open parentheses straight alpha plus straight beta close parentheses squared minus 2 αβ space space space space straight alpha squared plus straight beta squared equals 53 space space space space open parentheses straight alpha plus straight beta close parentheses squared minus 2 αβ equals 53 space space space space open parentheses negative straight k close parentheses squared minus 2 open parentheses negative open parentheses 2 straight k plus 4 close parentheses close parentheses equals 53 space space space space straight k squared plus 4 straight k plus 8 equals 53 space space space space straight k squared plus 4 straight k minus 45 equals 0 space space space space open parentheses straight k plus 9 close parentheses open parentheses straight k minus 5 close parentheses equals 0 space space space space straight k equals negative 9 space atau space straight k equals 5 end style$