Solving for X

Date: 5/14/96 at 23:44:24
From: Tanya Chongchit
Subject: X^2 - 7X + 10 = 0

I am having a totally hard time trying to figure out this equation and 
it is getting on my nerves!  Please help me and show me
each step to solving X.  Thank you!

          X^2 - 7X + 10 = 0

Is this what I am supposed to do?

          (X-5)(X-2)

Then is the answer X = 5 and X = 2?

Thank you again!  
Tanya

Date: 5/16/96 at 19:41:14
From: Doctor Syd
Subject: Re: X^2 - 7X + 10 = 0

Hey!  We're glad you wrote.  Sometimes it can certainly be frustrating 
when you don't think you know how to do a problem, I know!  But, hey, 
you got this problem!  

Do you understand why and how you factored the polynomial?  You were 
hoping that x^2 - 7x + 10 would factor into something nice, in the 
form (x + a)(x + b) where a and b are integers, right?  Well, how do 
you figure out what a and b are?  Let's expand the product:

(x + a)(x + b) = x^2 + (a+b)x + ab

We want to find out for what values of a and b we have that 
(x + a)(x + b) =  x^2 + (a-b)x + ab = x^2 -7x + 10

This means that we must have a-b = -7 and ab = 10.

You have two equations now and two unknowns.  You can either solve for 
a and b algebraically or you can try some numbers until you find some 
that work. You can see that a = -5 and b = -2 work, and thus you can 
say with confidence that

(x-2)(x-5)=x^2 - 7x +10

If this product is equal to 0, then one of the terms in the product is 
zero. In other words, either x-2 = 0 or x-5 =0.  Solving for x, this 
means that either x = 2 or x = 5.  Thus we have found the solution to 
the equation.

You had all of the right steps...I just filled in an explanation.  I 
hope this helps!  Good luck with other problems like this.  Once 
you've practice a little it will all seem much easier, I promise!

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