Misalkan m dan n adalah akar-akar persamaan kuadrat 3x2−5x+1=0 Persamaan kuadrat yang akar-akarnya 1+m21​dan 1+n21​ adalah....

Pertanyaan

Misalkan m dan n adalah akar-akar persamaan kuadrat 3 x squared minus 5 x plus 1 equals 0 Persamaan kuadrat yang akar-akarnya 1 plus 1 over m squareddan 1 plus 1 over n squared adalah....

  1. x squared minus 21 x minus 29 equals 0

  2. x squared minus 21 x plus 29 equals 0

  3. x squared plus 21 x plus 29 equals 0

  4. x squared minus 29 x minus 21 equals 0

  5. x squared plus 29 x plus 21 equals 0

M. Nasrullah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Makassar

Jawaban terverifikasi

Pembahasan

Ingat kembali penjumlahan dan perkalian pada fungsi kuadrat:

f open parentheses x close parentheses equals a x squared plus b x plus c

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell negative b over a end cell row cell x subscript 1 times x subscript 2 end cell equals cell c over a end cell end table  

Dan untuk persmaaan dengan akar-akar baru:

f open parentheses x close parentheses equals x squared minus open parentheses x subscript 1 plus x subscript 2 close parentheses x plus open parentheses x subscript 1 times x subscript 2 close parentheses 

Pada soal di dapat informasi:

3 x squared minus 5 x plus 1 equals 0 rightwards arrow a equals 3 comma space b equals negative 5 space dan space c equals 1

table attributes columnalign right center left columnspacing 0px end attributes row cell m plus n end cell equals cell negative b over a end cell row blank equals cell negative fraction numerator negative 5 over denominator 3 end fraction end cell row blank equals cell 5 over 3 end cell row blank blank blank row cell m. n end cell equals cell c over a end cell row blank equals cell 1 third end cell end table

Sehingga untuk persamaan baru yang akar-akarnya 1 plus 1 over m squareddan 1 plus 1 over n squared:

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell 1 plus 1 over m squared plus 1 plus 1 over n squared end cell row blank equals cell 2 plus 1 over m squared plus 1 over n squared end cell row blank equals cell 2 plus fraction numerator n squared plus n squared over denominator m squared times n squared end fraction end cell row blank equals cell 2 plus fraction numerator open parentheses n plus m close parentheses squared minus 2 m times n over denominator open parentheses m times n close parentheses squared end fraction end cell row blank equals cell 2 plus fraction numerator open parentheses begin display style 5 over 3 end style close parentheses squared minus 2 open parentheses begin display style 1 third end style close parentheses over denominator open parentheses begin display style 1 third end style close parentheses squared end fraction end cell row blank equals cell 2 plus fraction numerator begin display style 25 over 9 end style minus begin display style 2 over 3 end style over denominator begin display style 1 over 9 end style end fraction end cell row blank equals cell 2 plus fraction numerator begin display style fraction numerator 25 minus 6 over denominator 9 end fraction end style over denominator begin display style 1 over 9 end style end fraction end cell row blank equals 21 row blank blank blank row cell x subscript 1 times x subscript 2 end cell equals cell open parentheses 1 plus 1 over m squared close parentheses times open parentheses 1 plus 1 over n squared close parentheses end cell row blank equals cell 1 plus 1 over m squared plus 1 over n squared plus fraction numerator 1 over denominator m squared times n squared end fraction end cell row blank equals cell 1 plus fraction numerator n squared plus n squared over denominator m squared times n squared end fraction plus fraction numerator 1 over denominator m squared times n squared end fraction end cell row blank equals cell 1 plus fraction numerator open parentheses n plus m close parentheses squared minus 2 m times n plus 1 over denominator open parentheses m times n close parentheses squared end fraction end cell row blank equals cell 1 plus fraction numerator open parentheses begin display style 5 over 3 end style close parentheses squared minus 2 open parentheses begin display style 1 third end style close parentheses plus 1 over denominator open parentheses begin display style 1 third end style close parentheses squared end fraction end cell row blank equals cell 1 plus fraction numerator begin display style fraction numerator 25 minus 6 over denominator 9 end fraction plus fraction numerator begin display style 9 end style over denominator 9 end fraction end style over denominator begin display style 1 over 9 end style end fraction end cell row blank equals cell 1 plus 28 end cell row blank equals 29 end table

f open parentheses x close parentheses equals x squared minus open parentheses x subscript 1 plus x subscript 2 close parentheses x plus open parentheses x subscript 1 times x subscript 2 close parentheses f open parentheses x close parentheses equals x squared minus open parentheses 21 close parentheses x plus open parentheses 29 close parentheses f open parentheses x close parentheses equals x squared minus 21 x plus 29

Dengan demikian, persamaan kuadrat yang akar-akarnya 1 plus 1 over m squareddan 1 plus 1 over n squared adalah f open parentheses x close parentheses equals x squared minus 21 x plus 29

Jadi, jawaban yang tepat adalah B

 

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