Pertanyaan

Jika akar-akar persamaan kuadrat x 2 − 2 x − 6 = 0 adalah p dan q , maka persamaan kuadrat yang akar-akarnya dan p 2 1 + q 2 1 adalah

Jika akar-akar persamaan kuadrat $x^{2} - 2 x - 6 = 0$ adalah p dan q , maka persamaan kuadrat yang akar-akarnya dan $\frac{1}{p ^{2}} + \frac{1}{q ^{2}}$ adalah

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Klaim

F. Pratama

Master Teacher

Mahasiswa/Alumni Universitas Putra Indonesia YPTK Padang

Jawaban terverifikasi

Pembahasan

$begin mathsize 14px style Dari space persamaan space kuadrat space straight x squared minus 2 straight x minus 6 equals 0 space yang space akar minus akarnya space adalah space straight p space dan space straight q space diperoleh space straight p plus straight q equals negative fraction numerator left parenthesis negative 2 right parenthesis over denominator 1 end fraction equals 2 space space straight p space. straight q equals fraction numerator negative 6 over denominator 1 end fraction equals negative 6 space end style$

$begin mathsize 14px style Jadi space nilai space akar space persamaan space kuadrat space yang space baru colon straight x subscript 1 equals 1 plus straight p over straight q plus straight q over straight p equals fraction numerator pq plus straight p squared plus straight q squared over denominator pq end fraction straight x subscript 1 equals fraction numerator open parentheses straight p plus straight q close parentheses squared minus pq over denominator pq end fraction equals fraction numerator open parentheses 2 close parentheses squared minus open parentheses negative 6 close parentheses over denominator negative 6 end fraction straight x subscript 1 equals fraction numerator 4 plus 6 over denominator negative 6 end fraction equals negative 5 over 3 straight x subscript 2 equals 1 over straight p squared plus 1 over straight q squared equals fraction numerator straight q squared plus straight p squared over denominator straight p squared straight q squared end fraction straight x subscript 2 equals fraction numerator open parentheses straight p plus straight q close parentheses squared minus 2 pq over denominator open parentheses pq close parentheses squared end fraction equals fraction numerator open parentheses 2 close parentheses squared minus 2 left parenthesis negative 6 right parenthesis over denominator open parentheses negative 6 close parentheses squared end fraction straight x subscript 2 equals fraction numerator 4 plus 12 over denominator 36 end fraction equals 16 over 36 equals 4 over 9 maka space persamaan space kuadrat space baru space adalah open parentheses straight x minus open parentheses negative 5 over 3 close parentheses close parentheses open parentheses straight x minus 4 over 9 close parentheses equals 0 open parentheses straight x plus 5 over 3 close parentheses open parentheses straight x minus 4 over 9 close parentheses equals 0 straight x squared minus 4 over 9 straight x plus 5 over 3 straight x minus 20 over 27 equals 0 27 straight x squared minus 12 straight x plus 45 straight x minus 20 equals 0 27 straight x squared plus 33 straight x minus 20 equals 0 end style$