Question

MATERI : ALJABAR

Buktikan [tex]\displaystyle\rm x^2+\frac{1}{x^2}=3[/tex]


JIka diketahui x³ = 2+√5
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Answer 1

Jawaban:

Mencari nilai x:

[tex] \displaystyle \rm \large{x}^{2} + \frac{1}{ {x}^{2} } = 3[/tex]

[tex] \displaystyle \rm \large{x}^{2} + \frac{1}{ {x}^{2} } = 5 - 2[/tex]

[tex] \displaystyle \rm \large{x}^{2} + 2 + \frac{1}{ {x}^{2} } = 5[/tex]

[tex] \displaystyle \rm \large({x + \frac{1}{ x }})^{2} = 5[/tex]

[tex] \displaystyle \rm \large x + \frac{1}{x} = \sqrt{5} [/tex]

[tex] \displaystyle \rm \large x = \frac{ \sqrt{5} - 1 }{2},\:\frac{ \sqrt{5} + 1}{2} [/tex]

Uji coba nilai x:

untuk [tex] x = \frac{ \sqrt{5} - 1 }{2}[/tex]

[tex]\displaystyle \large \rm {(x)}^{3} = 2 + \sqrt{5} [/tex]

[tex]\displaystyle \large \rm {( \frac{ \sqrt{5} - 1 }{2} )}^{3} = 2 + \sqrt{5} [/tex]

[tex]\displaystyle \large \rm \sqrt{5 } - 2 = 2 + \sqrt{5}~~(tidak~memenuhi)[/tex]

untuk [tex]x = \frac{ \sqrt{5} + 1 }{2}[/tex]

[tex]\displaystyle \large \rm {( \frac{ \sqrt{5} + 1 }{2} )}^{3} = 2 + \sqrt{5} [/tex]

[tex]\displaystyle \large \rm2 + \sqrt{5} = 2 + \sqrt{5}~~(\bold{memenuhi})[/tex]

Sehingga:

[tex]\displaystyle \large \rm x = \frac{ \sqrt{5} + 1 }{2}[/tex]

[tex] \displaystyle \rm \large\frac{1}{x} = \frac{2}{ \sqrt{5} + 1} [/tex]

Maka:

[tex] \displaystyle \rm \large {x}^{2} + \frac{1}{ {x}^{2} } = 3[/tex]

[tex] \displaystyle \rm \large {x}^{2} + {(\frac{1}{ x})}^{2} = 3[/tex]

[tex] \displaystyle \rm \large{ (\frac{ \sqrt{5} + 1 }{2}) }^{2} + {(\frac{2}{ \sqrt{5} + 1 })}^{2} = 3[/tex]

[tex] \displaystyle \rm \large\frac{3 + \sqrt{5} }{2} + \frac{4 }{6+2\sqrt{5}} = 3[/tex]

[tex] \displaystyle \rm \large\frac{3 + \sqrt{5} }{2} + \frac{3 - \sqrt{5} }{2} = 3[/tex]

[tex] \displaystyle \rm \large \frac{3 + \sqrt{5} + 3 - \sqrt{5} }{2} = 3[/tex]

[tex] \displaystyle \rm \large\frac{6}{2} = 3[/tex]

[tex] \displaystyle \rm \large3 = 3[/tex]

Terbukti!

Answer 2

Penjelasan dengan langkah-langkah:

x² + 1/x² = 3

x² + 2 + 1/x³ = 5

x + 1/x = √5

x = √5 - 1/2

maka

x² +( 1/x)² = 3

3 - √5/2 + 4/6 - 2√5 = 3

6/2 = 3

3 = 3

terbukti