Operation # is defined as: a # b = a² + b² + 2ab, for all non-negative integers. What is the value of (a + b) – 7 when a # b = 196?

A. 5 B. 7 C. 11 D. 13 E. 25

(B) This problem is almost impossible to solve unless we reduce the expression first. From the given information we can safely conclude that: a # b = (a + b)². Now, if we solve the equation:

(a + b)² = 196 (a + b) = 14

Then (a + b) – 7 = 7. The answer is (B).

Notice, that we took the positive square root only since we were restricted to only non-negative integers (as stated in the question), otherwise we could have ended up with two solutions.

Can you please provide another solution to this problem. I do not understand the equations given in the answer.

Post subject: Re: MATH: Test 1, question5 : Symbols

Posted: Thu Apr 29, 2010 11:16 am

Joined: Fri Apr 09, 2010 2:11 pm Posts: 459

The "symbols questions" are the ones where we use some symbol to denote an operation.

In this case we define that a # b equals to a² + b² + 2ab. For any non-negative numbers a and b. (You can think of it as we substitute "a² + b² + 2ab" for "a # b").

Note that (-2) # 3 or (-4) # (-6) are not defined, because we stated that a and b are non-negative numbers.

We are told that a # b = 196. Let us write that in terms of common operations as a # b is defined:

a # b = a² + b² + 2ab = 196

a² + b² + 2ab = 196

So rewriting original questions statement:

We know that a² + b² + 2ab = 196. What is the value of (a + b) – 7 ?

We have two variables in the given equation, so we can not find a not knowing b and vice-versa. But we can see that

a² + b² + 2ab = (a + b)² .

It is a well-known formula. For reference here is how we calculate it: (a + b)² = (a + b) × (a + b) = a × a + a × b + b × a + b × b = a² + 2ab + b².

So we know that (a + b)² = 196

If we denote a + b as x then x² = 196.

x is 14 or -14 and so is a + b. But a + b can not equal to -14 because both a and b are non-negative numbers and therefore the sum of a + b is a non-negative number.

Since we know that a + b = 14 then (a + b) – 7 = 14 - 7 = 7.

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