中四聯合方程

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1. CE 2004/ll/42
If α ≠ β and
{ α² = 4α + 3
{ β² = 4β + 3,
then (α+1)(β+1) = ?

From the given information, α and β are two distinct roots of the quadratic equation x² = 4x + 3, or x² - 4x - 3 = 0.

Sum of roots = α + β = -(-4)/1 = 4
Product of roots = αβ = -3/1 = -3

Consider (α+1)(β+1) = αβ + α + β + 1
　　　　　　　　 = -3 + 4 + 1
　　　　　　　　 = 2

2. CE 2001/ll/39
If a, b are distinct real number and
{ a² + 4a + 1 = 0
{ b² + 4b + 1 = 0,
find a² + b².

From the given information, a and b are two distinct real roots of
x² + 4x + 1 = 0

Sum of roots = a + b = -4
Product of roots = ab = 1

Consider
a² + b² = a² + 2ab + b² - 2ab = (a + b)² - 2ab = (-4)² - 2(1) = 14