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Finding solutions to a quadratic equation by factoring

I'm going over my algebra so I can finally go to college. There's this one issue that I never really understood back in high-school:

When you have an quadratic equation, we were taught that you can solve it by using factoring.
My problem is that to factor, the book I have just tells you to *guess* two numbers that add up to equal x and multiply to equal y.

For example:
x*x - 11x + 28 = 0
can be converted to:

(x-4)(x-7) = 0
where you can solve it easily.

Isn't there a better way than guessing them? What would you do with seven-digit or non-integer numbers?

(When I tried solving the set of equations of:
m * n =  28
m + n = -11
I just got back to the original quadratic function- just with the m instead of x)

Don't mathematicians have a better method to do this? Math just always seemed the last place where guessing is the way to solve problems.

Monchanger
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Monchanger
Asked:
Monchanger
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1 Solution
 
Victor_RCommented:
What you are looking for is called the quadratic formula.

Use this to solve, for example: 4x^2 + 7x - 2 = 0

In this case, guessing is easy:
Adds to 7, multiplies to -8
so our numbers are 8 and -1
so we have (4x+8)(4x-1)=0
divide out the four (x+2)(x-1/4)=0
so our answers are -2 and 1/4

Now lets say we didn't know how to guess. How can we get our solution anyway?

Here's how:

4x^2 + 7x - 2 = 0  ---->    Ax^2 + Bx + C = 0

so A=4 B=7 and C=-2

x = -B +\- sqrt(B^2 - 4AC)
    -----------------------
              2A

x = -7 +\- sqrt[49-4(4)(-2)]
    -----------------------
              2(4)

solve this to find that x = 2/8 or x = -2

Which we know is the right answer because that's what we guessed.

If one of the answers becomes the root of a negative number, you say it's "inadmissible" (at least in high school).

http://www.sosmath.com/algebra/quadraticeq/quadraformula/quadraformula.html

Good luck!
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MonchangerAuthor Commented:
Thanks! And especially thanks for the URL. I love getting an explanation of how the formula works.

Now I only have to figure out a way to memorize that whole thing :-P

Monchanger
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Victor_RCommented:
Very welcome. Thanks for the A!

Let me know if you have more questions.
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