Question

If a and b are numbers such that (a − 4)(b + 6) = 0, then what is the smallest possible value of a^2 + b^2 ?

Collected in the board: Number Theory

Steven Zheng posted 6 hours ago

Answer

Since (a−4)(b+6) = 0, the possible solutions are:

a = 4 and b is anything, or b = −6 and a is anything.

Now, the expression a^2 + b^2 is made smallest by choosing a and b to be close to zero as possible. So, a = 4 and b = 0 will give us the smallest value of a^2 + b^2, namely, 16.

Using the other solution b = -6 and a = 0 would give the smallest value of 36, which will be bigger than 16.

Therefore

The smallest possible value of a^2 + b^2 is 16

Steven Zheng posted 6 hours ago

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