If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b

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Same answer. The highlighted portion would be 9a^2 - 96a?

Hi Richak91

Thanks for highlighting the typo error

Lets start solving this question.
Formula required : (x−y)(x+y)=(x2−y2)

\(\sqrt{3a}- \sqrt{2b + 2c} = 4\) ..............(i)

3a – 2b – 2c = 32
(\(\sqrt{3a}- \sqrt{2b + 2c}\))(\(\sqrt{3a}+ \sqrt{2b + 2c}\)) = 32
(4)(\(\sqrt{3a}+ \sqrt{2b + 2c}\)) = 32
(\(\sqrt{3a}+ \sqrt{2b + 2c}\)) = 8 ........................ (ii)

Adding (i) & (ii) , we get (\(2 \sqrt{3a}\)) = 12
a = 12

Subtracting (i) from (ii) , we get (\(2 \sqrt{2b + 2c}\)) = 4
b+c = 2

So, a+b+c = 12+2 = 14

Answer E

I will use plugging in . Given the equations and the options, a is likely 12 and a+b+c is 14. For further checking, 2b+2c = 4 and \sqrt{2b + 2c} = 2, making a=b=1. Hence, a+b+c = 12+2(1) = 14.

I don't think plugging in is a good idea for this question.... How did u directly arrived at a = 12 ??

Good question. It needs keen observation

1. From the first equation, 3a is likely to be greater than 32
2. From the second equation, [m]\sqrt{3a} is likely to be greater than 4

Put this inference together and a could be 12, i.e 3*12= 36>32, and \sqrt{3*12} = 6>4

The same could be used to know that a= b= 1, as explained in my earlier post. Even without doing this you should know by now that the answer is E.

This is a good must be true type of question, it could also make a decent data sufficiency question. Just observing keenly will obviously save a lot of time!


Another way to look at it is the form of the equation, take the first equation, 3a - 2a- 2b= 32. This can be rearranged as 3a - 2(a+b)= 32, therefore a must be an even number and a+b will also be even. Given this, a+b+c would be an even number of the x+2y. Only 10,12 and 14 are viable. Looking at the second equation will tell you that x/a would be 12 as 3a would form the only proper square.

3a>32 already includes [m]\sqrt{3a} is be greater than 4.
So we should further work on 3a>32..

But your observation is appreciable and surely can be used here to solve the problem using plugging method..

There is no need to actually work further once you know 3a>32, a is most likely 12 and the answer can only be 14, given the options.

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