Jawaban:

Beranda

Diketahui persamaan kuadrat 2x2−5x +1=0 mempunyai ...

Pertanyaan

Diketahui persamaan kuadrat begin mathsize 14px style 2 x squared minus 5 x space plus 1 equals 0 end style mempunyai akar-akar begin mathsize 14px style straight alpha end style dan begin mathsize 14px style beta end style. Persamaan kuadrat baru yang mempunyai akar-akar begin mathsize 14px style left parenthesis 2 straight alpha minus 1 right parenthesis end style dan size 14px left parenthesis size 14px 2 size 14px beta size 14px minus size 14px 1 size 14px right parenthesis adalah ....

begin mathsize 14px style x squared space minus space 3 x space minus space 2 space equals space 0 end style

size 14px x to the power of size 14px 2 size 14px space size 14px minus size 14px space size 14px 3 size 14px x size 14px space size 14px plus size 14px space size 14px 2 size 14px space size 14px equals size 14px space size 14px 0

size 14px x to the power of size 14px 2 size 14px space size 14px minus size 14px space size 14px 2 size 14px x size 14px space size 14px plus size 14px space size 14px 3 size 14px space size 14px equals size 14px space size 14px 0

size 14px x to the power of size 14px 2 size 14px space size 14px minus size 14px space size 14px 2 size 14px x size 14px space size 14px minus size 14px space size 14px 3 size 14px space size 14px equals size 14px space size 14px 0

size 14px x to the power of size 14px 2 size 14px space size 14px plus size 14px space size 14px 3 size 14px x size 14px minus size 14px space size 14px 2 size 14px space size 14px equals size 14px space size 14px 0

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I. Kumaralalita

I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

Pembahasan

Diketahui persamaan kuadrat begin mathsize 14px style 2 x squared minus 5 x space plus 1 equals 0 end style dengan akar-akar begin mathsize 14px style straight alpha end style dan begin mathsize 14px style beta end style, maka diperoleh

begin mathsize 14px style alpha plus beta equals negative b over a equals negative fraction numerator open parentheses negative 5 close parentheses over denominator 2 end fraction equals 5 over 2 alpha times beta equals c over a equals 1 half end style 

Misalkan akar-akar persamaan kuadrat yang baru adalah begin mathsize 14px style x subscript 1 space dan space x subscript 2 end style, dimana:

begin mathsize 14px style x subscript 1 equals 2 alpha minus 1 space dan space x subscript 2 equals 2 beta minus 1 end style.

sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell 2 alpha minus 1 plus 2 beta minus 1 end cell row blank equals cell 2 open parentheses alpha plus beta close parentheses minus 2 end cell row blank equals cell 2 open parentheses 5 over 2 close parentheses minus 2 end cell row blank equals cell 5 minus 2 end cell row blank equals 3 end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell x subscript 1 times x subscript 2 end cell equals cell open parentheses 2 alpha minus 1 close parentheses open parentheses 2 beta minus 1 close parentheses end cell row blank equals cell 4 alpha beta minus 2 alpha minus 2 beta plus 1 end cell row blank equals cell 4 alpha beta minus 2 open parentheses alpha plus beta close parentheses plus 1 end cell row blank equals cell 4 open parentheses 1 half close parentheses minus 2 open parentheses 5 over 2 close parentheses plus 1 end cell row blank equals cell 2 minus 5 plus 1 end cell row blank equals cell negative 2 end cell end table end style 

Dengan demikian, persamaan kuadrat yang baru menjadi:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell x squared minus open parentheses x subscript 1 plus x subscript 2 close parentheses x plus x subscript 1 x subscript 2 end cell equals 0 row cell x squared minus 3 x minus 2 end cell equals 0 end table end style