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Square root concept

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Square root concept

by vinni.k » Mon Oct 29, 2012 11:43 pm
This is a conceptual question.

I do understand this fact:-
x = √25
x=5

However, i have seen couple of examples which give different results. In one of the questions it gives one solution and in one of the questions it give two solutions.

I am marking this in bold. Please explain the difference between the two. Why one is giving only a single solution and other is giving two.

First Question:-
If f(x) = x^3 + √x and g(x) = 4x - 3, What is f(g(3)) ? (MGMAT -3 equations, inequalities & VIC, (functions strategy) Pg no. 73)
Solution as given in the book:-
g(3) = 4(3) - 3 = 9

f(g(3)) = f(9) = 9^3 + √9 = 729 + 3 = 732
This √9 is the biggest confusion only if i compare it with the below question.
Here, √9 = 3 and then added to 729. i.e 729 + 3 = 732.

Now, second question
If g(x) = 3x + √x , What is the value of g(d^2 + 6d + 9) ? (pg no. 79, Q2)
g(d^2 + 6d + 9) = 3(d^2 + 6d + 9) + √(d+3)^2

Now this one √(d+3)^2. How many solutions this must have ? According to me and if i compare it with the above question, it must have one solution i.e (d+3)

However, in the answer explanation there are two solutions + or - (d+3) which give two answers.
3d^2 + 19d + 30 or 3d^2 + 17d + 24


Please explain the difference between the two. I am answering questions incorrectly because of this concept.
Why there are two solutions for this √(d+3)^2 = + or - (d+3)

and one solution for this √9 = 3 or √3^2 = 3

Regards
Vinni
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by pemdas » Tue Oct 30, 2012 7:05 am
Any value squared is positive. We can obtain the square root of a value only if it's positive according to GMAT math convention. In general the square root of negative value is not real value, and it's complex value - depicted as having two dimensions on the number line one on x-abscess, the other on y-abscess.

Back to your immediate inquiry, we can obtain the square root only from a positive value. Since the value squared is positive, it's square root will be only positive value. But the attention must be paid to the value raised to even power(s). Likewise with squaring - we square value, even if it's negative and obtain the positive value.

There come two rules for you

square root (a^2) = a BUT a^2=(|a|)^2 implying that the value squared (raised to even power consequently) can be either positive value or negative, put the mode sign always.

Example:
square root (3^2) = square root (9) = 3 ALWAYS!
3^2=9=(-3)^2, square root (9) = 3 ALWAYS! BUT 9=(+-3)^2

In your question √(d+3)^2 we can set 9=(d+3)^2, then √9 = 3 ALWAYS! BUT (d+3)^2=9=|d+3| implying this could be d+3 or -d-3

to make your understanding easier, follow the sequence of operations. We first squared, then obtained the squared root. This implies that when we squared value, originally it was in the mod sign |d+3|, and of course after obtaining the square root we keep only the positive values i.e. (d+3). Because, we are interested in all (possible) solutions we have to follow the sequence of operations which means we could have -(d+3) too with the same answer obtained after obtaining square root of positive value (previously squared one).
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by vinni.k » Wed Oct 31, 2012 9:01 pm
Thank you so much Pemdas. Nice detailed explanation. :D

Regards
Vinni
pemdas wrote:Any value squared is positive. We can obtain the square root of a value only if it's positive according to GMAT math convention. In general the square root of negative value is not real value, and it's complex value - depicted as having two dimensions on the number line one on x-abscess, the other on y-abscess.

Back to your immediate inquiry, we can obtain the square root only from a positive value. Since the value squared is positive, it's square root will be only positive value. But the attention must be paid to the value raised to even power(s). Likewise with squaring - we square value, even if it's negative and obtain the positive value.

There come two rules for you

square root (a^2) = a BUT a^2=(|a|)^2 implying that the value squared (raised to even power consequently) can be either positive value or negative, put the mode sign always.

Example:
square root (3^2) = square root (9) = 3 ALWAYS!
3^2=9=(-3)^2, square root (9) = 3 ALWAYS! BUT 9=(+-3)^2

In your question √(d+3)^2 we can set 9=(d+3)^2, then √9 = 3 ALWAYS! BUT (d+3)^2=9=|d+3| implying this could be d+3 or -d-3

to make your understanding easier, follow the sequence of operations. We first squared, then obtained the squared root. This implies that when we squared value, originally it was in the mod sign |d+3|, and of course after obtaining the square root we keep only the positive values i.e. (d+3). Because, we are interested in all (possible) solutions we have to follow the sequence of operations which means we could have -(d+3) too with the same answer obtained after obtaining square root of positive value (previously squared one).
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