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Batas-batas nilai begin mathsize 14px style k end style agar size 14px y size 14px equals size 14px left parenthesis size 14px k size 14px minus size 14px 1 size 14px right parenthesis size 14px x to the power of size 14px 2 size 14px plus size 14px k size 14px x size 14px plus size 14px left parenthesis size 14px k size 14px minus size 14px 5 size 14px right parenthesis bernilai negatif untuk setiap begin mathsize 14px style x element of straight real numbers end style adalah .....

  1. begin mathsize 14px style k less than 1 end style

  2. undefined

  3. undefined

  4. begin mathsize 14px style k less than 4 minus fraction numerator 2 square root of 21 over denominator 3 end fraction end style atau  begin mathsize 14px style k greater than 4 plus fraction numerator 2 square root of 21 over denominator 3 end fraction end style

  5. undefined atau size 14px k size 14px less than size 14px 4 size 14px plus fraction numerator size 14px 2 square root of size 14px 21 over denominator size 14px 3 end fraction

H. Nufus

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

Dari begin mathsize 14px style left parenthesis k minus 1 right parenthesis x squared plus k x plus left parenthesis k minus 5 right parenthesis equals 0 end style, didapat begin mathsize 14px style a equals k minus 1 end stylebegin mathsize 14px style b equals k end style, dan begin mathsize 14px style c equals k minus 5 end style.

Ingat bahwa agar size 14px y size 14px equals size 14px left parenthesis size 14px k size 14px minus size 14px 1 size 14px right parenthesis size 14px x to the power of size 14px 2 size 14px plus size 14px k size 14px x size 14px plus size 14px left parenthesis size 14px k size 14px minus size 14px 5 size 14px right parenthesis bernilai negatif untuk setiap begin mathsize 14px style x element of straight real numbers end style, maka haruslah memenuhi syarat definit negatif, yaitu begin mathsize 14px style a less than 0 end style dan begin mathsize 14px style D less than 0 end style.

Syarat 1: undefined

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row a less than cell 0 space end cell row cell k minus 1 end cell less than cell 0 space end cell row k less than 1 end table end style

Syarat 2: undefined

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row D less than 0 row cell b squared minus 4 a c end cell less than 0 row cell k squared minus 4 left parenthesis k minus 1 right parenthesis left parenthesis k minus 5 right parenthesis end cell less than 0 row cell k squared minus 4 left parenthesis k squared minus 6 k plus 5 right parenthesis end cell less than 0 row cell k squared minus 4 k squared plus 24 k minus 20 end cell less than 0 row cell negative 3 k squared plus 24 k minus 20 end cell less than 0 end table end style

Terlebih dahulu, kita cari nilai begin mathsize 14px style k end style dari bentuk di atas dengan menggunakan rumus kuadratik.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell k subscript 1 , 2 end subscript end cell equals cell fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end cell row cell k subscript 1 , 2 end subscript end cell equals cell fraction numerator negative 24 plus-or-minus square root of 24 squared minus 4 left parenthesis negative 3 right parenthesis left parenthesis negative 20 right parenthesis end root over denominator 2 left parenthesis negative 3 right parenthesis end fraction end cell row cell k subscript 1 , 2 end subscript end cell equals cell fraction numerator negative 24 plus-or-minus square root of 576 minus 240 end root over denominator negative 6 end fraction end cell row cell k subscript 1 , 2 end subscript end cell equals cell fraction numerator negative 24 plus-or-minus square root of 336 over denominator negative 6 end fraction end cell row cell k subscript 1 , 2 end subscript end cell equals cell fraction numerator negative 24 plus-or-minus 4 square root of 21 over denominator negative 6 end fraction end cell row cell k subscript 1 , 2 end subscript end cell equals cell 4 minus-or-plus fraction numerator 2 square root of 21 over denominator 3 end fraction end cell end table end style

Didapat pembuat nolnya adalah begin mathsize 14px style k equals 4 minus fraction numerator 2 square root of 21 over denominator 3 end fraction end style atau begin mathsize 14px style k equals 4 plus fraction numerator 2 square root of 21 over denominator 3 end fraction end style.

Perhatikan garis bilangan berikut!

Karena tanda pertidaksamaannya adalah begin mathsize 14px style less than end style, maka pilih daerah yang bernilai negatif, yaitu undefined atau undefined.

Dari syarat begin mathsize 14px style a less than 0 end style dan begin mathsize 14px style D less than 0 end style, didapat batas-batas nilai begin mathsize 14px style k end style sebagai berikut.

begin mathsize 14px style k less than 1 end style

undefined

undefined

Oleh karena itu, didapat garis bilangan sebagai berikut.

Dengan demikian, batas-batas nilai begin mathsize 14px style k end style agar size 14px y size 14px equals size 14px left parenthesis size 14px k size 14px minus size 14px 1 size 14px right parenthesis size 14px x to the power of size 14px 2 size 14px plus size 14px k size 14px x size 14px plus size 14px left parenthesis size 14px k size 14px minus size 14px 5 size 14px right parenthesis bernilai negatif untuk setiap begin mathsize 14px style x element of straight real numbers end style adalah undefined.

Jadi, jawaban yang tepat adalah C.

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