SOLUTION: two numbers differ by 8, their product is 65,what is the number?

Question 705702: two numbers differ by 8, their product is 65,what is the number?
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
Some may quickly recognize that as a product, and being a factor of 5, 13*5=65, which we may also recognize as 12*5 plus one more 5. Basic addition fact may be in some people's heads and 8+5=13 is immediately recognized.

You ask for "what's the number", but the description indicates TWO numbers. I see 13 and 5.

You probably want an algebraic arithmetic treatment so here it is.

Numbers m and n.
Differ by 8: n-m=8
Product is 65: mn=65
Find m and n.

The Differ equation gives n=m+8. Use this in the Product equation.
m(m+8)=65
m^2+8m-65=0, which we can factor, so let's do that:
(m+5)(m-13) ---- will that work? No. Gives -8m instead of +8m.
(m-5)(m+13) ---- Will that work? Yes. 13m-5m=8m, and product in const. term is -65.

OKAY, solution to the quadratic formed is m=5 or m=-13.
What about n?
n-m=8
n=m+8
For m=5, n=13.
For m=-13, n=-5.

ANSWER: Both solutions fit the description. Either the numbers are -5 and -13, or they are 5 and 13.
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