An Incredible System of Equations | Math Olympiad
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- เผยแพร่เมื่อ 21 ส.ค. 2024
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In this algebraic video, we'll explore how to solve a challenging system of equations. Whether you're a math enthusiast or a student seeking to improve your problem-solving skills, these strategies will enhance your understanding and boost your confidence. We'll break down solution step-by-step, providing clear explanations and examples to ensure you grasp the concepts. Don't miss out on mastering these valuable techniques!
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System of equations
Algebra
Problem solving
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Factorization
Synthetic division
Quadratic Equation
Math enthusiast
Radical equations
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Rational root theorem
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Defining x^2/3=a and y^2/3=b, we get a^2+b^2=17 and a^3+b^3 = 65 > a=4, b=1 or a=1, b=4. If a=4, b=1 > (x,y) = (8,1), (-8,-1), (8, -1), (-8,1). If a=1, b=4, (x,y) = (1,8), (-1,-8), (1,-8), (-1,8).
(8 , 1). , (1 , 8)
Θετω (x^2)^(1/3)=α και (y^2)^(1/3)=β αρα εχω x^2=α^3 και y^2=β^3 α>0 και β>0. Αν (α+β)= t >0 εχωt^3-51t+130=0 t=5 αν t ανηκει στο Z.εχω τελικα α=4 β=1 ,ή α=1 β=4. Αρα x^2=64 y^2=1 ή x^2=1 y^2=64 και αρα x=+ -8 y=+ -1 ή x=+ -1 y=+ -8
x = u³
y = v³
u³u + v³v = 17
u⁴ + v⁴ = 17
u⁶ + v⁶ = 65
u² + v² = b
u²v² = c
t² - bt + c = 0
t² = bt - c
u⁴ = bu² - c
v⁴ = bv² - c
u⁴ + v⁴ = b(u² + v²) - 2c
17 = b² - 2c => c = (b² - 17)/2
t³ = bt² - ct = b(bt - c) - ct
t³ = (b² - c)t - bc
u⁶ = (b² - c)u² - bc
v⁶ = (b² - c)v² - bc
u⁶ + b⁶ = (b² - c)(u² + v²) - 2bc
65 = (b² - c)b - 2bc
65 = b(b² - 3c) = b(b² - 2c - c)
65 = b(17 - c)
65 = b[17 - (b² - 17)/2]
130 = b(34 - b² + 17)
130 = b(51 - b²)
b³ - 51b + 130 = 0
b³ - 125 - 51b + 255 = 0
b³ - 5³ - 51(b - 5) = 0
(b - 5)(b² + 5b - 26) = 0
b - 5 = 0 => b = 5
c = (b² - 17)/2 => c = 4
u² + v² = 5
u²v² = 4
p² - 5p + 4 = 0
(p - 1)(p - 4) = 0
u² = 1 => v² = 4
x = u³ => x = ± 1
y = v³ => y = ± 8
u² = 4 => v² = 1
x = u³ => x = ± 8
y = v³ => y = ± 1
(x, y) =
{ (1, 8), (1, -8), (-1, 8), (-1, -8),
(8, 1), (8, -1), (-8, 1), (-8, -1) }
(x,y)=(1,8); (-1,-8); (8,1); (-8,-1)
By inspection!
@@TheDavidlloydjones I did it this way also. Inspection should precede long manipulations of symbols. One has to get a sense of a problem before heading into a blizzard of identities.