A Nice Math Olympiad Problem | What Is The Value Of 'a' and 'b' In This Problem??? | Fast Trick!

This image shows a simple algebraic system of two equations:

1. a + b= 11

2. a.b = 111,

with the goal of solving for the unknown variables A and B

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From equation (1) we get a = 11-b. Insert this into (2) to get (11-b)b = 111 -> 11b - b² = 111 -> b²-11b-111 = 0. Now solve for b1, 2 by applying the quadratic formula to get b1 = 33/2 and b2 = -11/2. Insert either solution into equation (1) to find that in case of b1, a1 would become -11/2, while in case of b2, a2 would calculate as 33/2. Hence there are two reciprocal solutions for real numbers of a and b: [a1, b1] = [-11/2, 33/2] and [a2, b2] = [33/2, -11/2].