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Diketahui bentuk persamaan kuadrat (m−1)x2−(3m+1)x+4m=0. Tentukan nilai m sedemikian rupa sehingga jumlah kuadrat akar-akarnya sama dengan 13!

Pertanyaan

Diketahui bentuk persamaan kuadrat open parentheses m minus 1 close parentheses x squared minus open parentheses 3 m plus 1 close parentheses x plus 4 m equals 0.

Tentukan nilai m sedemikian rupa sehingga jumlah kuadrat akar-akarnya sama dengan 13!

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

nilai m yang memenuhi adalah m equals 1 third space logical or space m equals 3.

Pembahasan

Dari persamaan open parentheses m minus 1 close parentheses x squared minus open parentheses 3 m plus 1 close parentheses x plus 4 m equals 0, diperoleh:

a equals m minus 1 comma space b equals negative open parentheses 3 m plus 1 close parentheses comma space c equals 4 m 

Jika alpha comma space beta maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell alpha plus beta end cell equals cell negative b over a end cell row blank equals cell negative fraction numerator open square brackets negative open parentheses 3 m plus 1 close parentheses close square brackets over denominator m minus 1 end fraction end cell row blank equals cell fraction numerator 3 m plus 1 over denominator m minus 1 end fraction end cell row cell alpha beta end cell equals cell c over a end cell row blank equals cell fraction numerator 4 m over denominator m minus 1 end fraction end cell end table 

Menentukan nilai m sehingga jumlah kuadrat akar-akarnya sama dengan 13:

table attributes columnalign right center left columnspacing 0px end attributes row cell alpha squared plus beta squared end cell equals 13 row cell open parentheses alpha plus beta close parentheses squared minus 2 alpha beta end cell equals 13 row cell open parentheses fraction numerator 3 m plus 1 over denominator m minus 1 end fraction close parentheses squared minus 2 open parentheses fraction numerator 4 m over denominator m minus 1 end fraction close parentheses end cell equals 13 row cell open parentheses 3 m plus 1 close parentheses squared over open parentheses m minus 1 close parentheses squared minus fraction numerator 8 m over denominator m minus 1 end fraction end cell equals 13 row cell fraction numerator open parentheses 3 m plus 1 close parentheses open parentheses 3 m plus 1 close parentheses over denominator open parentheses m minus 1 close parentheses open parentheses m minus 1 close parentheses end fraction minus fraction numerator 8 m open parentheses m minus 1 close parentheses over denominator open parentheses m minus 1 close parentheses open parentheses m minus 1 close parentheses end fraction end cell equals 13 row cell fraction numerator 9 m squared plus 6 m plus 1 over denominator m squared minus 2 m plus 1 end fraction minus fraction numerator 8 m squared minus 8 m over denominator m squared minus 2 m plus 1 end fraction end cell equals 13 row cell fraction numerator m squared plus 14 m plus 1 over denominator m squared minus 2 m plus 1 end fraction end cell equals 13 row cell 13 open parentheses m squared minus 2 m plus 1 close parentheses end cell equals cell m squared plus 14 m plus 1 end cell row cell 13 m squared minus 26 m plus 13 end cell equals cell m squared plus 14 m plus 1 end cell row cell 12 m squared minus 40 m plus 12 end cell equals 0 row cell 3 m squared minus 10 m plus 3 end cell equals 0 row cell 3 m squared minus 9 m minus m plus 3 end cell equals 0 row cell 3 m open parentheses m minus 3 close parentheses minus open parentheses m minus 3 close parentheses end cell equals 0 row cell open parentheses 3 m minus 1 close parentheses open parentheses m minus 3 close parentheses end cell equals 0 row cell 3 m minus 1 end cell equals 0 row cell 3 m end cell equals 1 row m equals cell 1 third end cell row cell m minus 3 end cell equals 0 row m equals 3 end table  

Jadi, nilai m yang memenuhi adalah m equals 1 third space logical or space m equals 3.

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