Soal dan Pembahasan Olimpiade Matematika Vektor Nasional (OMVN) 2012 Tingkat SMP Nomor 7 (Babak Penyisihan)

Soal:

Untuk bilangan real positip x, jika x – 12 = ¹/_x maka nilai dari  x⁴ – 10x³ –15x² –122x + 2012 = ...

Pembahasan:

Diketahui: x – 12 = ¹/_x

x – 12 = ¹/_x         ==> x² – 12x = 1                  (dikalikan x)

                          ==> x² = 12x + 1

x – 12 = ¹/_x         ==> x³ – 12x² = x                 (dikalikan x²)

                          ==> x³ = 12x² + x

                          ==> x³ = 12(12x + 1) + x

                          ==> x³ = 144x + 12 + x

                          ==> x³ = 145x + 12

x – 12 = ¹/_x         ==> x⁴ – 12x³ = x²                 (dikalikan x³)

                          ==> x⁴ = 12(145x + 12) + (12x + 1)

                          ==> x⁴ = 1740x + 144 + 12x + 1

                          ==> x⁴ = 1752x + 145

Dengan demikian:

x⁴ – 10x³ –15x² – 122x + 2012    = 1752x + 145 – 10(145x + 12) – 15(12x + 1) – 122x + 2012

                                                    = 1752x + 145 – 1450x – 120 – 180x – 15 – 122x + 2012

                                                    = 1752x – 1752x + 10 + 2012

                                                    = 2022

Jadi, nilai dari  x⁴ – 10x³ –15x² –122x + 2012 = 2022

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Pelajari Juga: [Nomor 1], [Nomor 2], [Nomor 3], [Nomor 4], [Nomor 5], [Nomor 6], 
[Nomor 8], [Nomor 9], [Nomor 10], dan [Nomor 11 - 20]