If x = ¾ and y = ⅖ , What is the Value of \(\sqrt{(x^2+ 6x+ 9)}\) - \(\sqrt{(y^2-2y+ 1)}\)?

Question: If x = ¾ and y = ⅖ , what is the value of \(\sqrt{(x^2+ 6x+ 9)}\) - \(\sqrt{(y^2-2y+ 1) }\)?

1. 87/20
2. 63/20
3. 47/20
4. 15/4
5. 14/5

This topic is a part of GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "Manhattan Prep's "GMAT Foundations of Math, 5th Edition"" published in the year 2007. GMAT Quant section consists a total of 31 questions. This is a GMAT Problem Solving question that allows the candidates to select the correct answer from the five answer choices provided. The total time allotted for this part is 62 minutes which allows the candidate 2 minutes to answer each question.

Solution and Explanation:

Approach Solution 1

Let us write down the original equation:

\(\sqrt{(x^2+ 6x+ 9)} - \sqrt{(y^2-2y+ 1)}\)

Using the properties:

a^2 + b^2 + 2ab = (a+b)^2 and

a^2 + b^2 - 2ab = (a-b)^2

We can formulate the original equation with respect to the above properties:

\(\sqrt{(x^2+ 6x+ 9)} - \sqrt{(y^2-2y+ 1)} = \sqrt{(x^2+2*x*3+ 3^2)} - \sqrt{(x^2-2*y*1+ 1^2)}\)

Therefore, we will get:

\(\sqrt{(x^2+ 6x+ 9)} - \sqrt{(y^2-2y+ 1)} = \sqrt{(x+3)^2 - (y- 1)^2}\)

\(\sqrt{(x+3)^2} - \sqrt{(y- 1)^2} = |x+3| - |y-1|\)

Put the values of x and y from the given question:

|x+3| - |y-1| = |¾ +3 | - |⅖ -1 |
=15/4 - 3/5
=63/20

Hence, the correct option is B.

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