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Nie obliczając pierwiastków wyznaccz wartość wyrażenia
x² + 3x - 4=0
x₁⁴ + x₂⁴=
Wyrazenie to doprowadzam do postaci wzorów Viete`a :
x₁ + x₂ = -b/a
x₁*x₂ = c/a
a = 1
b= 3
c= -4
x₁⁴ + x₂⁴= (x₁² +x₂²)² -2x¹*x²=
= [( x₁ +x₂)² - 2x₁x₂]² - 2*(x₁x₂)²=
= [ ( -b/a)² - 2*c/a]² - 2*(c/a)² =
= [ (-3/1)² -2*(-4/1)]² -2*(-4/1)² =
= [ 9 + 8 ]² - 2*(-4)²=
= [ 17]² - 2*16 =
= 289 - 32 =
= 257
Sprawdzam poprzez wyliczenie pierwiastków
x² + 3x - 4=0
Δ = 3² - 4*1*(-4) = 9 + 16 = 25
√Δ = √25 = 5
x1 = (-3-5): 2*1 = (-8) : 2 = -4
x2 = (-3+5): 2*1 = (2 ) : 2 = 1
x₁⁴ + x₂⁴= (- 4)⁴ + 1⁴ = 256 + 1 = 257