Solving Equations that Equal Zero

Date: 02/26/97 at 21:58:17
From: Thomas G. Morton, Jr.
Subject: Help with Algebra

Dear Dr. Math:

My son is in the 8th grade.  I am trying to help him with his algebra 
but I really need some pointers. 

For example:

(1) (5x+10)(7x-2) = 0  How do I solve it for x?

(2) 3x(2x-1)(x+2) = 0  In solving for x, are there 3 answers or 2?

(3) How do I find the solution set of 2x^2 - 4x + 2 = 0? 

These are problems my son missed on his test. I can't solve them 
either. Any ideas would really be appreciated. 

Date: 02/26/97 at 23:33:35
From: Doctor Mike
Subject: Re: Help with Algebra

Hi Mr. Morton,
  
There is a key fact that should help you here.  If a product of 2 
numbers (2 numbers multiplied together) is equal to zero, then at 
least one of them must be zero.  In the case of your first problem 
this means that either 5x+10 = 0 or 7x-2 = 0.  The solutions to these 
2 equations are x = -2 and x = 2/7, respectively, so these are the 
solutions to the original equation.  Your second problem can be solved 
in a similar fashion.

For problems like (3) where the equation is not already factored as in 
your first 2 problems, it is best to make the equation as simple as 
possible before we start to factor.  In this case, since all the terms 
on the left side have an even number as their coefficient, we can 
factor 2 out of everything to get: 2(x^2-2x+1)=0  

This factors as 2(x-1)(x-1)=0  

Since both of these factors are the same there is only one answer, 
namely x = 1.
  
Factoring expressions like this is not a simple thing.  There are some 
techniques which are often useful, but some of these expressions 
cannot be factored no matter how hard you try (unless you go to 
"complex numbers" which I think you can put off for a while). There is 
not an easy way to pick this up. It just needs to be studied and 
practiced. If you encounter another specific exercise that stubbornly 
resists your efforts to figure it out, please write back and we may be 
able to help.

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/