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### 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
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/
```
Associated Topics:
Middle School Factoring Expressions

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