Selamat Bekerja!.... Question from @Shiqo

December 2018 1 202 Report

$\star\ \bigstar \ \boxed{\boxed{\boxed{\bold{Brain\ Tester}}}}\ \bigstar\ \star\\ Materi : Teori Bilangan (atau Persamaan kuadrat) Jenjang: SMP Level : Sedang\\ Dilarang menjawab Spam, barang siapa menjawab spam, maka dia termasuk orang-orang yang merugi.\\ Pertanyaannya:\\ [tex]\d{Diberikan\ bilangan\ real $a,\ $b yang memenuhi} \\ \\ \d{$a^{2} = -101a + \sqrt{2017} } \\ \d{$b^{2} = 101b + \sqrt{2017}} \\ \\ \d{Tentukan nilai Kuadrat dari hasil kali bilangan itu!}$

Selamat Bekerja!

whongaliem a² = - 101.a + √2017
a² + 101.a = √2017

$a^{2} + 101.a + \frac{ 101^{2} }{4} = \frac{ 101^{2} }{4} + \sqrt{2017}$

$(a + \frac{101}{2}) ^{2} = \frac{ 101^{2} + 4 \sqrt{2017} }{4}$

$a + \frac{101}{2} = \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }$

$a = - \frac{101}{2} + \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }$

$a^{2} = [ - \frac{101}{2} + \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} } ]^{2}$

$a^{2} = \frac{ 101^{2} }{4} - 101. \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} } + \frac{ 101^{2} + 4 \sqrt{17} }{4}$

$a^{2} = \frac{ 101^{2} }{2} + \sqrt{2017} - 101\sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }$

b² = 101.b + √2017
b² - 101.b = √2017

$b^{2} - 101.b + \frac{ 101^{2} }{4} = \frac{ 101^{2} }{4} + \sqrt{2017}$

$(b - \frac{101}{2}) ^{2} = \frac{ 101^{2} + 4 \sqrt{2017} }{4}$

$b - \frac{101}{2} = \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }$

$b = \frac{101}{2} + \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }$

$b^{2} = [ \frac{101}{2} + \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }] ^{2}$

$b^{2} = \frac{ 101^{2} }{4} + 101. \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} } + \frac{ 101^{2} + 4 \sqrt{2017} }{4}$

$b^{2} = \frac{ 101^{2} }{2} + \sqrt{2017} + 101 \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }$

(a.b)² = a² .b²

= $[(\frac{ 101^{2} }{2} + \sqrt{2017} ) - 101. \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} } ]$ $[( \frac{ 101^{2} }{2} + \sqrt{2017} ) + 101 . \sqrt{ \frac{ 101^{2} + 4 \sqrt{2017} }{4} }$

= $( \frac{ 101^{2} }{2} + \sqrt{2017}) ^{2} - (110. \sqrt{ \frac{ 110^{2} + 4 \sqrt{2017} }{4} }) ^{2}$

= $\frac{ 101^{4} }{4} + 101^{2} \sqrt{2017} + 2017 - [ 101^{2} ( \frac{ 101^{2} + 4 \sqrt{2017} }{4} )$

= $\frac{ 101^{4} }{4} + \sqrt{2017} + 2017 - [ \frac{ 101^{4} + 4 \sqrt{2017} }{4} )$

= $\frac{ 101^{4} }{4} + \sqrt{2017} + 2017 - \frac{ 101^{4} }{4} - \sqrt{2017}$

= 2017

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