Question 203192
How far apart are consecutive even integers? In other words, how much to you add to one even integer to get to the next one? I hope you understand that the answer is two. So if we call out smaller positive even integer "x", then the next one up would be "x+2".<br>
Now that we have expressions for the two consecutive even integers we are in a position to translate "the positive difference between the squares of two consecutive positive even integers is 52"
"The positive difference": Difference means subtraction and a subtraction that works out positive means the first number must be larger than the first.
"the squares of two consecutive positive even integers": {{{x^2}}} and {{{(x + 2)^2}}}
"The positive difference between the squares of two consecutive positive even integers": {{{(x + 2)^2 - x^2}}}
"The positive difference between the squares of two consecutive positive even integers is 52": {{{(x + 2)^2 - x^2 = 52}}}<br>
Now we have an equation to solve. First we simplify:
{{{(x + 2)^2 - x^2 = 52}}}
{{{x^2 + 4x + 4 - x^2 = 52}}}
{{{4x + 4 = 52}}}
This is a simple equation to solve. Subtract 4 from both sides:
{{{4x = 48}}}
Divide both sides by 4:
{{{x = 12}}}
Since x was the smaller positive even integer, the other one is 2 more or 14.<br>
Checking our work:
{{{14^2 - 12^2 = 52}}}
{{{196 - 144 = 52}}}
{{{52 = 52}}} Check!