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Persamaan kuadrat dengan p > 0, mempunyai akar-akar dan . Jika mempunyai akar-akar dan , maka q - p = ...

Persamaan kuadrat begin mathsize 12px style 2 x squared minus p x plus 1 equals 0 end style dengan p > 0, mempunyai akar-akar begin mathsize 12px style alpha end style dan begin mathsize 12px style beta end style. Jika begin mathsize 12px style x squared minus 5 x plus q equals 0 end style mempunyai akar-akar begin mathsize 12px style 1 over alpha squared end style dan size 12px 1 over size 12px beta to the power of size 12px 2, maka q - p = ...

  1. - 2

  2. begin mathsize 12px style negative 1 half end style

  3. begin mathsize 12px style 1 half end style

  4. 1

  5. 2

Jawaban:

Perhatikan sifat persamaan kuadrat

begin mathsize 12px style 2 x squared minus p x plus 1 equals 0  a equals 2 comma space b equals negative p comma space c equals 1  alpha plus beta equals negative b over a equals negative fraction numerator open parentheses negative p close parentheses over denominator 2 end fraction equals p over 2 space midline horizontal ellipsis space open parentheses 1 close parentheses  alpha times beta equals c over a equals 1 half space midline horizontal ellipsis space open parentheses 2 close parentheses end style

Misalkan begin mathsize 12px style 1 over alpha squared equals m end style dan size 12px 1 over size 12px beta to the power of size 12px 2 size 12px equals size 12px n, maka:

begin mathsize 12px style m plus n equals 1 over alpha squared plus 1 over beta squared  m plus n equals fraction numerator alpha squared plus beta squared over denominator alpha squared times beta squared end fraction  m plus n equals fraction numerator open parentheses alpha plus beta close parentheses squared minus 2 alpha times beta over denominator alpha squared times beta squared end fraction space midline horizontal ellipsis space open parentheses 3 close parentheses end style

 

Sebustitusikan pers (1) dan (2) ke pers (3)

begin mathsize 12px style fraction numerator open parentheses begin display style p over 2 end style close parentheses squared minus 2 times begin display style 1 half end style over denominator open parentheses 1 half close parentheses squared end fraction  equals fraction numerator begin display style fraction numerator begin display style p squared end style over denominator begin display style 4 end style end fraction minus 1 end style over denominator begin display style fraction numerator begin display style 1 end style over denominator begin display style 4 end style end fraction end style end fraction  equals fraction numerator begin display style fraction numerator begin display style p squared end style over denominator begin display style 4 end style end fraction minus fraction numerator begin display style 4 end style over denominator begin display style 4 end style end fraction end style over denominator begin display style fraction numerator begin display style 1 end style over denominator begin display style 4 end style end fraction end style end fraction  equals fraction numerator begin display style p squared minus 4 end style over denominator begin display style 4 end style end fraction times 4 over 1  equals p squared minus 4    m times n equals 1 over alpha squared times fraction numerator begin display style 1 end style over denominator begin display style beta squared end style end fraction equals fraction numerator begin display style 1 end style over denominator begin display style open parentheses alpha beta close parentheses squared end style end fraction equals fraction numerator 1 over denominator begin display style open parentheses begin display style 1 half end style close parentheses squared end style end fraction equals 4 end style

 

Maka persamaan kuadrat yang akar-akarnya begin mathsize 12px style 1 over alpha squared end style dan size 12px 1 over size 12px beta to the power of size 12px 2 yaitu:

begin mathsize 12px style x squared minus open parentheses m plus n close parentheses x plus m times n equals 0 end style, menjadi:

begin mathsize 12px style x squared minus open parentheses p squared minus 4 close parentheses x plus 4 equals 0 end style, yang ekuivalen dengan undefined

Maka

begin mathsize 12px style p squared minus 4 equals 5  p squared equals 9  p equals 3 space d a n space q equals 4 end style

Maka q - p = 4 - 3 = 1

0

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