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Pertanyaan

Persamaan kuadrat 2x2 - px + 1 = 0 dengan p > 0 , mempunyai akar-akar α dan ß. Jika x2 - 5x + q = 0 mempunyai akar-akar begin mathsize 14px style 1 over alpha squared end style dan begin mathsize 14px style 1 over beta squared end style maka q — p = ....

  1. -2

  2. begin mathsize 14px style 1 half end style

  3. begin mathsize 14px style 1 half end style

  4. 1

  5. 2

N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

2x2 - px + 1 = 0 akar-akarnya α dan ß

α + ß = begin mathsize 14px style negative b over a equals p over 2 end style

α . ß = begin mathsize 14px style c over a equals 1 half end style

x2 - 5x + q = 0 akar-akarnya begin mathsize 14px style 1 over alpha squared end style dan begin mathsize 14px style 1 over beta squared end style

begin mathsize 14px style 1 over alpha squared end style + begin mathsize 14px style 1 over beta squared end style = begin mathsize 14px style negative b over a end style

begin mathsize 14px style fraction numerator alpha squared plus beta squared over denominator alpha squared beta squared end fraction end style = 5

begin mathsize 14px style fraction numerator open parentheses alpha plus beta close parentheses squared minus 2 alpha beta over denominator open parentheses alpha beta close parentheses squared end fraction end style = 5

begin mathsize 14px style fraction numerator open parentheses p over 2 close parentheses squared minus 2 open parentheses 1 half close parentheses over denominator open parentheses 1 half close parentheses squared end fraction end style = 5

begin mathsize 14px style p squared over 4 end style - 1 = begin mathsize 14px style 5 over 4 end style

p2 - 4 = 5
p2 = 9
p = ± 3
Karena p > 0, maka yang diambil p = 3

begin mathsize 14px style 1 over alpha squared end style . begin mathsize 14px style 1 over beta squared end style = begin mathsize 14px style c over a end style

begin mathsize 14px style fraction numerator 1 over denominator alpha squared beta squared end fraction end style = q

begin mathsize 14px style fraction numerator 1 over denominator 1 fourth end fraction end style = q

q = 4

q - p = 4 - 3 = 1

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