Question 117950
The two unknown integers are represented by x and by x+1
.
If you square the two integers you get:
.
{{{x^2}}} 
and
{{{(x+1)^2}}}
.
Squaring {{{(x+1)}}} results in {{{(x+1)^2 = (x+1)*(x+1) = x^2 + 2x + 1}}}
.
If you subtract the two squares, the problem tells you that the result is 25. In equation form
this subtraction can be expressed as:
.
{{{x^2 + 2x +1 - x^2 = 25}}}
.
Notice that in the subtraction the two {{{x^2}}} terms are of opposite sign and therefore
cancel each other. Therefore, you are left with:
.
{{{2x + 1 = 25}}}
.
Get rid of the 1 on the left side by subtracting 1 from both sides to get:
.
{{{2x = 24}}}
.
Solve for x by dividing both sides of this equation by 2 to reduce the equation to:
.
{{{x = 24/2 = 12}}}
.
So one of the unknown integers (that is x) equals 12. The other integer is x + 1 and therefore
it equals 12 + 1 or 13.
.
The two integers you are looking for are 12 and 13.
.
Check. 12 squared is 144 and 13 squared is 169. The difference is 169 - 144 = 25, just as the
problem requires. So this answer checks and the two integers are 12 and 13.
.
Hope this helps you to understand the problem.
.