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#1 2005-11-26 06:09:14

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Factorising Quadratics

I was revising some chapters of my A level maths, I was wondering;
When factorising a quadratic where the coefficient of the x^2 is more than 1, is there a way of figuring out 'a' and 'b' where (ax + y) (bx + z) without using trial and error?
i.e. 6x^2 - 11x + 3
= (2x - 3) (3x - 1)

I start of by guessing with (6x + _) (x + _) and try to fill in the blanks, if it dosent work I move on to (2x + _) (3x + _) and so on.  If there is a quicker way to determin the coefficients of x, it would be tres handy smile

Last edited by rickyoswaldiow (2005-11-26 06:09:58)


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#2 2005-11-26 07:31:17

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Factorising Quadratics

The best way to describe the method is just to give an example.
Let's do it with your one.

6x² - 11x + 3 = 0

The way to solve this is similar to solving ones where the x² coefficient is 1, but instead of finding 2 numbers that add to give -11 and multiply to give 3, we need numbers that add to give -11 and multiply to give 6x3 = 18, because the x² coefficient is 6.

Two such numbers are -9 and -2.
We can use these numbers to rewrite the original equation in a different form:
6x² - 9x - 2x + 3 = 0

Factorising the first two terms gives: 3x(2x - 3) - 2x + 3 = 0
Factorising the last two terms gives: 3x(2x - 3) -1(2x - 3) = 0 [1 included for clarity]

These two terms will combine to give (2x-3)(3x-1) = 0
The two x coefficients will always factorise with the other 2 terms to make the overall factorisation easier, no matter what coefficients you start off with. (Provided, of course, that they factorise in the first place)

If you're doing A level, you probably know this already, but you can check if a quadratic will factorise by looking at the discriminant: b² - 4ac, where a, b and c come from ax² + bx + c = 0.

For the quadratic to factorise, the discriminant has to be a square number.


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#3 2005-11-26 07:59:23

RickyOswaldIOW
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Registered: 2005-11-18
Posts: 212

Re: Factorising Quadratics

so if we have ax^2 + bx + c we need to find 2 numbers that add to give b and multiply to give a*c?

Last edited by rickyoswaldiow (2005-11-26 08:00:51)


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#4 2005-11-26 08:59:54

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Factorising Quadratics

Exactly. You can then use those two numbers to split up the b and that helps you factorise the quadratic easier.


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#5 2006-01-23 02:53:55

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Factorising Quadratics

I am stuck on the question y=-2x^2 - 7x + 15 where I need to sketch the curve.
x = 0 => y = 15

-2x^2 - 7x + 15
-[2x^2 + 7x - 15]
   Here, I can see that the two numbers I need are +10 and -3 since:
10 * -3 = 2 * -15
   and
10 - 3 = 7
  so would the answer be...


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#6 2006-01-23 02:58:13

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Factorising Quadratics

(ax + b) (cx + d)
so
-2x^2 - 7x + 15
-[2x^2 + 7x - 15]
-[(2x - 3)(x + 5)]
my problem is removing the square set of brackets here.

Last edited by rickyoswaldiow (2006-01-23 03:09:42)


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#7 2006-01-23 05:54:29

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Factorising Quadratics

solved

(-2x + 3)(x + 5) or (2x - 3)(-x - 5)


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#8 2006-01-23 08:55:49

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Factorising Quadratics

(-2x+3)(x+5)

I wouldn't bother trying to get that negative sign out of this problem, as you can see, it only served to confuse you.  Mathsyperson's detailed description is all that you need.

-2x² - 7x + 15  (you realized that -10 and +3 were the factors needed)

-2x² - 10x + 3x + 15

-2x(x + 5) + 3(x + 5)

(x + 5)(-2x + 3)

  I am actually thrilled to be able to do this myself.  I hadn't seen, or did not remember, this method until Mathsyperson layed the steps out perfectly.

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#9 2006-02-12 07:00:35

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Factorising Quadratics

I'm still having quite a bit of confusion over finding the co-efficients of the x in (dx + e) (fx + g).  I can follow your examples and do other similar sums myself but I want to know an exact way of getting those co-efficients for my C++ based computer program.

for(i=0; i<255; i++){
   for(j=0; j<255; j++){
     while(a*c==i*j & b==i+j){ cout << "(x + " << i << ")(x + " << j) };
   }
}

a, b and c are doubles entered by the user, how do I expand this program to take in values of a (ax^2 + bx + c) that are not 1 and then display the co-efficients of x in the answer?


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#10 2006-02-12 07:49:32

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Factorising Quadratics

In finding the factors of ax² + bx + c = 0

ac = n

d + e = b

d = n / e

d = ac / e

b = ac / e  + e 

b = (ac + e²) / e

be = ac + e²

e² - be + ac = 0

e = (b ± √(b² - 4ac)) / 2a

The two solutions for e actually represent both factors i.e.,  d and e

This definition will lead you to;

ax² + dx + ex + c

But now you would need to factorize;

ax² + dx   and also  ex + c

I'll have to think about this one some more.


edit*

  It seems that after you had a, d, e, and c,  you would then need an expression which would find the greatest common factor of a and d  and then the gcf of e and c.

  If f1 were the greatest factor of a and d  and  f2 were the gfc of e and c;

  Your solution would be;

  (f1 + f2) (a/f1 + d/f1)

  If (a/f1 + d/f1) ≠ (e/f2 + c/f2), then it is not factorizable.... I think.

  I have to leave this for someone smarter than me from this point.  Sorry I couldn't be more helpful.

Last edited by irspow (2006-02-12 08:47:36)

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#11 2006-02-12 10:05:18

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Factorising Quadratics

You could try something like this to find f1 or f2.

  Declare a variable n and then make an iterative loop where n starts at the smaller value of the two numbers in question.  For each iteration is would decrease by 1.  Then declare a variable f1 (greatest common denominator)
 
  In the loop make a statement so that if;


  a%n = a/n  and  d%n = d/n    then    f1 = n

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#12 2006-02-14 03:34:18

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Factorising Quadratics

what does the % operator do? wink


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#13 2006-02-14 08:55:29

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Factorising Quadratics

It's been a while, but I believe that in C++ it is the modulus(?) function.  It only returns an integer value for a fraction (or division).  What the statement up there would do is check whether it is true that n goes evenly into a and d.  If it didn't go in evenly then the integer answer wouldn't match the float answer.  Being false, it would not assign f1 = n.

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#14 2006-02-14 09:19:40

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Factorising Quadratics

% returns the remainder of division:

5 % 2 = 1
10 % 3 = 1
100 % 10 = 0
5 % 5 = 0
22 % 4 = 2

If you wish to see whether n goes into a (n divides a), you would do:

if (a % n == 0)


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#15 2006-02-14 09:45:00

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Factorising Quadratics

Thanks Ricky, it's been a while.

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#16 2006-02-14 11:04:41

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Factorising Quadratics

Yep.  The syntax is a bit weird becaue there is no mathimatical equivalent (that I know of).  The only thing close is:

a = x mod n

Where x would be the result from a % n.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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