Square and Square roots

While preparing for the CAT, you have probably come across a discussion that says x^2 = 4 has two roots, 2 and -2, while √4 has only one value, 2. Now, let’s try to understand why this is so: 1) x^2 = 4 Basic algebra tells us that quadratics have two roots. Here, x can be either 2 or -2; each, when squared, will give you 4. x^2 – 4 = 0 and (x + 2)*(x – 2) = 0 when x equals -2 or 2. 2) √x is positive, only Now this is odd, right? √4 must be 2. Why is that? Shouldn’t it be 2 or -2. After all, when we square both 2 and -2, we get 4 (as discussed above). So, √4 should be 2 or -2. Here is the concept: √x denotes only the principal square root. x has two square roots – the positive square root (or principal square root) written as √x and the negative square root written as -√x. Therefore, when you take the square root of 4, you get two roots: √4 and -√4, which  is 2 and -2 respectively. On a CAT question, when you see √x, this is specifically referring to the positive square root of the number. So √4 is 2, only. 3) (√x)^2 = x This is fairly straightforward – since x has a square root, it must be non-negative. When you square it, just the square root sign vanishes and you are left with x. 4) √(x^2) = |x| Now this isn’t intuitive either. √(x^2) should simply be x – why do we have absolute value of x, then? Again, this has to do with the principal square root concept. First you will square x, and then when you write √, it is by default just the principal square root. The negative square root will be written as -√(x^2). So, irrespective of whether x was positive or negative initially, √(x^2) will definitely be positive x. Therefore, we will need to take the absolute value of x. Here’s a quick recap with some examples:

• √9 = 3
• x^2 = 16 means x is either 4 or -4
• √(5^2) = 5
• √(-5^2) = 5
• (√16)^2 = 16
• √100 = 10

To see this concept in action, let’s take a look at a very simple official problem: If x is not 0, then √(x^2)/x = (A) -1 (B) 0  (C) 1 (D) |x|/x We know that √(x^2) is not simply x, but rather |x|. So, √(x^2)/x = |x|/x. Depending on whether x is positive or negative, |x|/x will be 1 or -1 – we can’t say which one. Hence, there is no further simplification that we can do, and our answer must be D. Now that you are all warmed up, let’s examine a higher-level question: Is √[(x – 3)^2] = (3 – x)? Statement 1: x is not 3 Statement 2: -x * |x| > 0 We know that √(x^2) = |x|, so √[(x – 3)^2] = |x – 3|. This means that our question is basically: Is |x – 3| = 3 – x? Note that 3 – x can also be written as -(x – 3). Is |x – 3| = -(x – 3)? Recall the definition of absolute values: |a| = a if a is greater than or equal to 0, and -a if a < 0. So, “Is |x – 3| = -(x – 3)?” depends on whether (x – 3) is positive or negative. If (x – 3) is negative (or 0), then |x – 3| is equal to -(x – 3). So our question now boils down to: Is (x – 3) negative (or 0)? Statement 1: x is not 3 This means we know that (x – 3) is not 0, but we still don’t know whether it is negative or positive. This statement is not sufficient. Statement 2: -x * |x| > 0 |x| is always non-negative, so for the product to be positive, “-x” must also be positive. This means x must be negative. If x is negative, x – 3 must be negative, too. If (x – 3) is negative, |x – 3| is equal to -(x – 3). Hence, this statement alone is sufficient, and our answer is B.