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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10, SAT-II.

What is the minimum of the expression:

equation

(by C. Ionescu-Tiu)

Notice that, by adding a constant number, we can force a perfect square out of:

equation

equation

and similarly for b.

Therefore, subtracting back what we added in order to force the perfect squares:

equation

equation

Since the minimum value of any square is zero, the expression always satisfies:

equation

Therefore, the minimum value of the expression E is 4.5.