1) x² - 10x + q = 0
a = 1 b = - 10 c = q
X₁ = 4 X₂
будем пользоваться теоремой Виета:
X₁ + X₂ = - b
X₁ * X₂ = c (q)
тогда 4X₂ + X₂ = - (-10) 5X₂ = 10 X₂ = 10/5 X₂ = 2, тогда X₁ = 4X₂ = 4*2 = 8
X₁*X₂=c(q) 8*2 = 16 Ответ : q = 16
2) x² + 7x+3 = 0
D = b² - 4 ac = 49 - 12 = 37
X₁ = (-b + √D) / 2a = (-7 +√37) / 2
X₂ = (- 7 - √D) / 2
тогда X₁² + X₂² =( (-7+√37) / 2)² + ((-7 - √37) / 2)² =
((49 - 14√37 + 37) / 4) + ((49+14√37 +37) /4) = (86 - 14√37 + 86 +14√37)/4 = 172/4 = 43
ответ X₁² + X₂² = 43
3) x² - 10 x + 3 = 0
D = 100 - 12 = 88
X₁ = (10 + √88) / 2
X₂ = (10 - √88) / 2
X₁³ + X₂³ = (X₁+X₂)*(X₁² - X₁*X₂ + X₂²)
(X₁+X₂) = (10 + √88) / 2 + (10 - √88) / 2 = (10+√88+10 - √88)/2 = 20/2 = 10
X₁² = ((10+√88) / 2)² = (100+20√88+88) / 4 = (188 + 20√88) / 4 = (4(47+5√88)) / 4 = 47+5√88
X₁*X₂ = ((10+√88) / 2) * ((10 - √88) / 2) = (100-88) / 4 = 12/4 = 3
X₂² = ((10-√88) / 2)² = (100-20√88 + 88)/4 = (4(47-5√88))/4 = 47-5√88
теперь подставим все в исходную формулу
X₁³ + X₂³ = (X₁+X₂)*(X₁² - X₁*X₂ + X₂²) = (10) * ((47+5√88) - (3) + (47-5√88)) = 10 * (47+5√88 - 3 + 47 - 5√88) = 10*(97) = 970
ответ : X₁³ + X₂³ = 970