Coefficients 4445

Find all triplets

P (x) = a * x^{2} + b * x + c

with the integer coefficients a, b, and c to which it applies
P (1)

while
((P (1))² + ((P (2))² + ((P (3))² = 22.

Correct answer:

a₁ = -2
b₁ = 11
c₁ = -12
a₂ = 2
b₂ = -5
c₂ = 0

Step-by-step explanation:

a_{1} = - 2 b_{1} = 11 c_{1} = - 12 x_{1} = 1 x_{2} = 2 x_{3} = 3 P_{1} = a_{1} \cdot x_{1}^{2} + b_{1} \cdot x_{1} + c_{1} = (- 2) \cdot 1^{2} + 11 \cdot 1 + (- 12) = - 3 P_{2} = a_{1} \cdot x_{2}^{2} + b_{1} \cdot x_{2} + c_{1} = (- 2) \cdot 2^{2} + 11 \cdot 2 + (- 12) = 2 P_{3} = a_{1} \cdot x_{3}^{2} + b_{1} \cdot x_{3} + c_{1} = (- 2) \cdot 3^{2} + 11 \cdot 3 + (- 12) = 3 P_{1} < P_{2} < P_{3} d = P_{1}^{2} + P_{2}^{2} + P_{3}^{2} = (- 3)^{2} + 2^{2} + 3^{2} = 22

b_{1} = 11

c_{1} = (- 12) = - 12

a_{2} = 2 b_{2} = - 5 c_{2} = 0 P_{1} = a_{2} \cdot x_{1}^{2} + b_{2} \cdot x_{1} + c_{2} = 2 \cdot 1^{2} + (- 5) \cdot 1 + 0 = - 3 P_{2} = a_{2} \cdot x_{2}^{2} + b_{2} \cdot x_{2} + c_{2} = 2 \cdot 2^{2} + (- 5) \cdot 2 + 0 = - 2 P_{3} = a_{2} \cdot x_{3}^{2} + b_{2} \cdot x_{3} + c_{2} = 2 \cdot 3^{2} + (- 5) \cdot 3 + 0 = 3 P_{1} < P_{2} < P_{3} d = P_{1}^{2} + P_{2}^{2} + P_{3}^{2} = (- 3)^{2} + (- 2)^{2} + 3^{2} = 22

b_{2} = (- 5) = - 5

c_{2} = 0

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Mathematical Olympiad

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