Lets call our integers x and y.
How do we represent the "sum of their squares" with a mathematical expression?
This would be x2 + y2 and we are told this is equal to 145.
But I can't solve one equation with two variables. x2 + y2 = 145 has many solutions. The trick is in how you represent "consecutive integers." They come one after the other such as (4,5), (26,27) etc. How do we get from one number to the next consecutive one? We add one. So if my first number is x, the next consecutive number is x + 1. So lets replace the y in our equation with (x+1)
x2 + (x+1)2 = 145
Now we have an equation with one variable that I can solve.
Foil first to get x2 +x2 +2x + 1 = 145
2x2 + 2x +1 = 145
Since this is a quadratic equation we can use the quadratic formula or factor to solve it.
First get everything to one side 2x2 + 2x - 144 = 0
and you will get that x = 8 or -9
These are the two possible answers for the first consecutive integer x. Then we can figure out the next consecutive integer (y) by adding one to each. So there are two sets of answers, 8 and 9, or -9 and -8. A simple check shows they both will give you 145.