Question 790572: A girl is 3 years younger than her brother. The product of their ages is 238. Find their ages.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The easiest way to the answer (in my opinion) is to factor 238.
Dividing by 2, I find that 
119 does not divide evenly by 2, 3, or 5, but it divides by 7:
So, 
The sister is  and the brother is  .
If you are studying prime factorization, or if you are not in advanced algebra, that is the way you are expected to solve the problem.
If you are studying quadratic equations, you may be expected to write and solve one, in a long, painful, unnecessary waste of effort, like this:
 = age of the brother
 = age of the sister
That is the quadratic equation.
There are 3 popular ways to solve such an equation.
You could use factoring, "completing the square", or the quadratic formula.
Factoring involves finding two factors that multiply to yield 238, and differ by 3. In a way, that is what I did above, without even mentioning algebra or equations.
Factoring, we find that  so we re-write the equation as
At that point we realize that a solution must make  or 
--> which cannot be the brother's age, because it is a negative number.
leads to  = age of the brother, and
 = age of the sister.
Completing the square would be:
So either  -->  (which cannot be the age of the brother), or
 -->  = age of the brother,
so = age of the sister.
Using the quadratic formula:
The solutions to  can be expressed as
In the case of ,
 ,  , and  , so
So the solutions to that equation are:
 (which is not a solution to the problem, because the brother's age cannot be negative),
and  = age of the brother,
so = age of the sister.
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