Persamaan mempunyai akar-akar dan . Dengan demikian maka ...

Pertanyaan

Persamaan $2 to the power of 2 x end exponent minus 3 times 2 to the power of x plus 2 end exponent plus 32 equals 0$ mempunyai akar-akar $x subscript 1$ dan $x subscript 2$ . Dengan demikian maka $x subscript 1 plus x subscript 2 equals$ ...

32 $space$
12 $space$
7 $space$
5 $space$
2 $space$

Pembahasan Soal:

Diketahui:

Persamaan $2 to the power of 2 x end exponent minus 3 times 2 to the power of x plus 2 end exponent plus 32 equals 0$

Ditanya:

$x subscript 1 plus x subscript 2$

Bentuk $a p to the power of 2 x end exponent plus b p to the power of x plus c equals 0$ merupakan persamaan eksponen yang dapat direduksi menjadi persamaan kuadrat.

Perlu diingat bahwa:

$k to the power of open parentheses l plus m close parentheses end exponent equals k to the power of l times k to the power of m open parentheses k to the power of l close parentheses to the power of m equals k to the power of l times m end exponent$

Perhatikan perhitungan berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell 2 to the power of 2 x end exponent minus 3 times 2 to the power of x plus 2 end exponent plus 32 end cell equals 0 row cell 2 to the power of 2 x end exponent minus 3 times 2 to the power of x times 2 squared plus 32 end cell equals 0 row cell 2 to the power of 2 x end exponent minus 3 times 2 to the power of x times 4 plus 32 end cell equals 0 row cell 2 to the power of 2 x end exponent minus 12 times 2 to the power of x plus 32 end cell equals 0 row cell 2 to the power of 2 x end exponent minus 12 times 2 to the power of x plus 32 end cell equals 0 end table$

Misalkan $2 to the power of x$ adalah $a$ , maka:

$space space space 2 to the power of 2 x end exponent minus 12 times 2 to the power of x plus 32 equals 0 open parentheses 2 to the power of x close parentheses squared minus 12 times 2 to the power of x plus 32 equals 0 space space space space space space space space a squared minus 12 a plus 32 equals 0 space space space space space space space space space open parentheses a minus 8 close parentheses open parentheses a minus 4 close parentheses equals 0 space space space space a minus 8 equals 0 space logical or space a minus 4 equals 0 space space space space space space space space space space a equals 8 space logical or space a equals 4$

Karena $a equals 2 to the power of x$ sehingga, $x subscript 1$ dan $x subscript 2$ dapat ditentukan seperti berikut:

$space space a equals 8 space space space space space atau space space space space space a equals 4 2 to the power of x subscript 1 end exponent equals 8 space space space space space space space space space space space space space space 2 to the power of x subscript 2 end exponent equals 4 2 to the power of x subscript 1 end exponent equals 2 cubed space space space space space space space space space space space space space 2 to the power of x subscript 2 end exponent equals 2 squared space x subscript 1 equals 3 space space space space space space space space space space space space space space space space x subscript 2 equals 2$

Hasil penjumlahan akar-akar $x subscript 1$ dan $x subscript 2$ :

$table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 end cell equals cell 3 plus 2 end cell row blank equals 5 end table$

Sehingga, hasil penjumlahan akar-akar $x subscript 1 plus x subscript 2$ adalah $5$ .

Jadi, jawaban yang tepat adalah D.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 19 Juli 2021

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