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.
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).
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!