How do you factor quadratic expressions?

Question

I do not understand this AT ALL. Please explain anything and everything you know about this topic please.

Answer 1

Suppose you have a quadratic equation in the form:    Ax² + Bx + C = 0

 

What has to happen for this equation to have two factors in the form:

                                                    (Dx + E)(Fx + G)

Using FOIL,

       We know immediately that    A = D*F

                           and that          C = E*G

 

                and that  (D*G + E*F) = B

 

That means we look at B and C as the sum (considering negative numbers) and the product of numbers.  It usually is very helpful to find all of the factors of B and C if we are dealing with easy algebra problems.  Here's an example:

 

                    Factor:      x² + 3x -10 =

                                    (x + 5)(x-2) = 0

                                      x = -5   or  x = 2

That problem is solved easily if you list the factors of 10:

                                             10 has factors of 1, 2, 5

Any of those factors have a sum (or difference) of -3 ?      Yes,   -5 + 2

            So, put those factors into  (Dx + E)(Fx + G)

                                         D=1

                                         F=1

                                         E = -5

                                         G = 2

 

Now, when D and F are not equal to 1, this gets a little more complicated.  See if you can solve this problem for A, B, and C in this case.  It still has products and differences, so that what you continue to look for when factoring.

 

It gets easier with practice.

Answer 2

Quadratic expression means a sum of terms (each term has form ax^n , where a is a constant (real number), and n is an integer, such as 0, 1, 2, etc.), and x stands for a variable (x, y, z, p, etc.)(the same variable across the whole expression!).

Quadratic means that one term in the expression has n=2 (such as the term: 3x^2), and there are no terms with n>2 present.

So some typical quadratic expressions might be:

3x^2 + 2x - 1

x^2 - 2

22y^2 + 234y + 222444567

Note that a constant term (example: - 3) can be thought of as -3x^0.

Note also that a quadratic expression doesn't have to have all three of the usual terms -- it might be missing the "x" term, or it might be missing the constant term. Each of these cases has its own, quite simple solution for factoring:

x^2 - 4 = (x-2)(x+2) and x^2 - 2x = (x)(x-2), for example.

Factor means to find a way of breaking down the quadratic expression into the product of two terms, each of which has 1 as its highest power of the variable.

So as examples, a factored expression might be written as (3x - 2)(2x + 3).

YOUR job is:

1) to let your brain clue in on patterns (such as successive signs on the terms of the expression being: +, -, +) and start to form the possible factors with this in mind -- a little though and calculation should convince you that obtaining that expression requires factors of form (ax - b)(cx - d), where a, b, c, d are all positive numbers).

2) usually your courses will require you to find only factors with integer coefficients -- that's the a, b, c, and d things. So you need to use just a little basic math to pick some possible sets of values for a and c, and b and d, and see how the product comes out. (a*c must equal the x^2 term's coefficient, and b*d must equal the constant).

As you get better at trying out values -- by which I mean, if you first get an expression with the coefficients that's far away from the desired expression, you'll be able to pick your next trial set of (a and c), (b and d) so as to get either the desired expression, of one that is much closer to the desired expression -- this will all come easier to you.

Hope this helps you, a little.

Answer 3

To factor these types of equations, we use FOIL. This method factors using two binomials.  Suppose you have the following equation.

 

x² + 4x + 3

 

 

We factor it as

 

(x + 3)(x + 1)