Help with a math question!!

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1

<p>Hello everyone,</p>

<p>I've come across this question doing an old official practice test (with no answer explanations). I get .5 but the answer key says 2.5, and I have no idea why.</p>

<p>Question: If 0 <= x <= y and (x+y)^2 - (x-y)^2 >= 25, what is the least possible value of y?</p>

<p>My thinking:
- 2 squares that are 25 apart: 144 and 169
- thus x+y=13 and x-y=12
- subtracting one equation by the other gives y=0.5</p>

<p>Can someone tell me what I'm doing wrong?</p>

<p>Any help would be greatly appreciated!</p>

2

<p>I would solve it this way (I could be wrong though):</p>

<p>(x+y)^2 - (x-y)^2 => 4xy
so 4xy >=25</p>

<p>Since you are solving for the least possible value of y,
4xy = 25
xy = 6.25</p>

<p>since x <= y, the least possible value can be 2.5</p>

3

<p>@jeff0309, your solution is incorrect. Who said that squares of numbers have to be integers? Also, x is less than y.</p>

<p>I would do what happyORmom did. Alternatively, you can apply difference of two squares.</p>

4

<p>Ohh I see now… Thank you both!</p>

5

<p>If you want a strictly “algebraic” method, follow this approach: (x + y)^2 - (x - y)^2 ≥ 25 = 4xy ≥ 25 = y ≥ 25/(4x). Since we want the LEAST, simply set x = to it. x = 25/(4y). Since 0 ≤ x ≤ y, plug in to get: 0 ≤ 25/(4x) ≤ y. Solve and get that y ≥ 5/2. Hence, the least value of y is 5/2.</p>