Does a^2 + b^2 = 2ab?

I will go with E

because ...
(1)
(a+b)^2 = a^2 + 2ab + b^2
9 = a^2 + b^2 + 2ab

different values will satisfy this equation..
not sufficient

(2) ab > 0
so what? not sufficient

(1) + (2)
even 2 statements together do not help much...

Simplying the question stem,
\(a^2 + b^2 - 2ab = 0\) -> \(a^2 + b^2 + 2ab - 4ab = 0\) -> \((a + b)^2 - 4ab = 0\)

The information given in either statement is not enough separately, but when we combine the
information from both the statements,

the question stem becomes \(9 - 4ab = 0\) where ab > 0

However, 2 cases are possible

Case 1: ab = 2.25, then the equation is true!
Case 2: ab = 3, the equation is not true (Insufficient - Option E)

Target question: Does a² + b²= 2ab?

This is a good candidate for REPHRASING the target question.
Take a² + b²= 2ab and subtract 2ab from both sides to get: a² - 2ab + b²= 0
Factor to get: (a - b)(a - b) = 0
Solve to get: a - b = 0
In other words, a = b

So, we get....
REPHRASED target question: Does a = b?

NOTE: Once we REPHRASE the target question, the statements should be easy to analyze.

Statement 1: a + b = 3
There are several values of a and b that satisfy statement 1. Here are two:
Case a: a = 1 and b = 2. In this case, the answer to the REPHRASED target question is NO, a does NOT equal b
Case b: a = 1.5 and b = 1.5. In this case, the answer to the REPHRASED target question is YES, a DOES equal b
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: ab > 0
There are several values of a and b that satisfy statement 2. Here are two:
Case a: a = 1 and b = 2 (these values satisfy the condition that ab > 0). In this case, the answer to the REPHRASED target question is NO, a does NOT equal b
Case b: a = 1.5 and b = 1.5. In this case, the answer to the REPHRASED target question is YES, a DOES equal b
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: a = 1 and b = 2. In this case, the answer to the REPHRASED target question is NO, a does NOT equal b
Case b: a = 1.5 and b = 1.5. In this case, the answer to the REPHRASED target question is YES, a DOES equal b
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

RELATED VIDEO

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.

We can modify the question in the following way.

\(a^2 + b^2 = 2ab\)
\(⇔ a^2 - 2ab + b^2 = 0\)
\(⇔ (a-b)^2 = 0\)
\(⇔ a = b\)

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) :
\(a = 1.5\), \(b = 1.5\) : The answer is Yes
\(a = 1\), \(b = 2\): The answer is No.

Since we don't have a unique solution, both conditions together are not sufficient.

Therefore, the answer is E.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

Inequalities have certain properties such as :

a^2 + b^2 = 2ab ; when a=b

So, we need to see if any option helps us establish that a=b

OPTION 1 :

a+b = 3

First look might lead you to believe that they are not equal. You might think of options such as :

1+2 = 3
2+1 = 3
0+3 = 3 and so on..

Please note that the equation DOES NOT say that 'a' and 'b' are integers. So, they can be equal :

1.5 + 1.5 = 3

Since there can be multiple values of 'a' and 'b', this option is INSUFFICIENT

OPTION 2:

ab > 0

This simply tells us that the product is positive. 'a' and 'b' can take any number of values. Hence, this option is also INSUFFICIENT.

COMBINING option 1 and option 2 also does not establish 'a=b'

Therefore, OPTION E is the right answer.

Hi Bunuel, do you have a list of questions that relate to (a+b)^2, (a-b)^2, and a^2-b^2 ?

7. Algebra

• Theory
  Math : Algebra 101
  Algebra: Tips and hints
  FOIL on the GMAT: Simplifying and Expanding
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  Using Algebra vs. Logic on GMAT Quant Questions
  
  • Questions
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    Algebra Ability Quiz by e-GMAT
    DS Questions
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For more check Ultimate GMAT Quantitative Megathread

Hope it helps.