Question 33038
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(1) Following the directions to solve the problem using factoring....<br>
Let the smaller integer be x; then the larger integer is x+2.  The sum of the squares is 340, and the integers are negative:<br>
{{{x^2+(x+2)^2=340}}}
{{{x^2+x^2+4x+4=340}}}
{{{2x^2+4x-336=0}}}
{{{x^2+2x-168=0}}}
{{{(x+14)(x-12)=0}}}
{{{x=-14}}} or {{{x=12}}}<br>
The integers are negative, so the smaller integer is x = -14 and the large integer is x+2 = -12.<br>
ANSWERS: -14 and -12<br>
(2) Being smart about how you use algebra....<br>
Use the powerful "trick" shown by tutor @ikleyn -- instead of using x and x+2 for the two integers, use x-1 and x+1.  Then<br>
{{{(x-1)^2+(x+1)^2=340}}}
{{{x^2-2x+1+x^2+2x+1=340}}}
{{{2x^2=338}}}
{{{x^2=169}}}
{{{x=13}}} or {{{x=-13}}}<br>
The answers have to be negative, so x is -13 and the two integers are x-1 = -14 and x+1 = -12.<br>
As you can see, using this trick leads to an equation that is easily solved and  does not require the use of factoring.  That's the reason for using the trick (in this and similar problems).<br>
(3) Solving the problem as quickly as possible -- as if you are taking a timed competitive exam.<br>
Solve informally using logical reasoning and mental arithmetic.<br>
Half of 340 is 170.<br>
What are the two squares of even integers that are closest to and on opposite sides of 170? They are 12^2 = 144 and 14^2 = 196.<br>
The answers have to be negative, so they are -14 and -12.<br>