Penjumlahan dan Perkalian Akar-Akar Persamaan Kuadrat

23 Februari 11

Hem, kali ini saya begitu jenuh dengan banyak sekali tugas kantor yang belum terselesaikan. Di sela-sela kesibukan ini saya mencoba untuk berbagi apa yang ada di dalam otak saya saat ini. Materi ini sering keluar dalam ujian akhir nasional tingkat SMA, yaitu mencari hasil penjumlahan atau perkalian atau pembagian akar-akar sebuah persamaan kuadrat. Dalam materi ini kadang kala bervariasi dengan mencari konstanta dari persamaan kuadrat tersebut.
Bentuk baku dari persamaan kuadrat adalah [pmath]ax^2+bx+c=0[/pmath]. Sedangkan contoh soal yang sesuai dengan judul di atas adalah sebagai berikut:
Jika [pmath] alpha dan beta [/pmath] adalah akar-akar persamaan kuadrat [pmath]x^2-8x+12=0[/pmath], maka nilai dari [pmath]alpha+beta [/pmath] adalah….
Untuk menyelesaikn soal tersebut ada dua cara :

Cara I : Mencari akar-akar persamaannya lalu menjumlahkan

[pmath]doubleleftright x^2-8x+12=0[/pmath]
[pmath]doubleleftright (x-2)(x-6)=0[/pmath]
[pmath]doubleleftright x-2=0 atau x-6=0[/pmath]
[pmath]doubleleftright x=2 atau x=6[/pmath]
Jadi , [pmath]alpha = 2 dan beta=6[/pmath], maka [pmath]alpha+beta=2+6=8[/pmath]

Cara II: dengan rumus langsung

Dari persamaan kuadrat [pmath]x^2-8x+12=0[/pmath] diketahui a=1, b=-8 dan c=12.
Maka: [pmath]alpha+beta=-(b/a)=-(-8/1)=8[/pmath], nah sama bukan dengan jawaban cara I.
Sedangkan rumus untuk mecari perkalian dua buah akarnya adalah [pmath]c/a[/pmath]. Itu aja dulu deh, lain kali akan saya sambung dengan variasi soal yang lain.

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