<p>For some reason I don't understand this one... argh.</p>
<p>If a and b are positive integers and a^2 - b^2 = 7, what is the value of a?</p>
<p>A. 3
B. 4
C. 5
D. 6
E. 7</p>
<p>Eh.</p>
<p>Alright. Since these are squares, you need to write down the perfect squares you know...</p>
<p>1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25....and etc...</p>
<p>now that you know them.. Try to look for two numbers thathave a difference of 7. only 3 and 4. But Since this is a positive number... the larger number should be a!!!</p>
<p>hope this helped</p>
<p>I thought I could approach it this way... but I figured there would be a faster solution, and a way to do it algebraically. So I guess Collegeboard was expecting you to solve it that way, by basically guessing which numbers it could be?</p>
<p>The fastest solution is better than any "proper" solution. If you want to solve every problem under 30 seconds, you'll have to become expert at exploring all cases.</p>
<p>Algebraically:
You should recognize that the left hand side factors into (a-b)(a+b) and since 7 is prime, the only resulting possibilities indicate that a must be 4</p>
<p>^wouldn't have thought of it like that in a million years.</p>
<p>i just thought:</p>
<p>1
4
9
16
25</p>
<p>25 - x = 7; x = 18, which doesn't work
16 - x = 7; x = 9, which means your answer is 4</p>
<p>seems sort of elementary, but it works and it's fast. well under 30 seconds.</p>