How do you factor an equation?

Question

Please show me an example and explain it carefully, I need to learn by tomorrow, cause I have my math exam.

Answer 1

There is no single way that can be explained here. It takes a few weeks of class to teach it.

Answer 2

There are three examples that are very helpful to memorize:

DIFFERENCE OF TWO SQUARES
(a^2 - b^2) = 0

e.g.
(x^2 - 9)
(y^2 - z^2)
(x^2 - 11)

These equations factor into the form (a + b)(a - b), always. The above examples, therefore, become:

(x + 3)(x - 3)
(y + z)(y - z)
(x + sqrt(11))(x - sqrt(11))

DIFFERENCE OF TWO CUBES
(a^3 - b^3) = 0

e.g.
(x^3 - 64)
(z^3 - 8)

They always factor according to the following rule:
(a - b)(a^2 + ab + b^2)

Examples:
(x - 4)(x^2 + 8x + 16)
(z - 2)(z^2 + 2z + 4)

SUM OF TWO CUBES
(a^3 + b^3) = 0

e.g.
(x^3 + 27)
(x^3 + 1)

Very similarly, they factor according to the following rule:

(a + b)(a^2 - ab + b^2)

So the above examples then become:
(x + 3)(x^2 - 3x + 9)
(x + 1)(x^2 - x + 1)

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Sometimes, however, you'll get a quadratic equation such as the following:

x^2 - 5x - 6 = 0

In this circumstance, you'll need to break it down so you start like this:

(x )(x )

You need to find two numbers that, when multiplied, give you the constant term (here, -6), and when added together, give you the coefficient of the linear term (here, -5)

Pairs of numbers that give you -6 when multiplied:
(-1, 6), (-2, 3), (-3, 2), (-6, 1)

Of these, pairs that add to -5:
(-6, 1)

Once you find a pair such as this, those two numbers are the coefficients in the factors you have set up above:

(x - 6)(x + 1)

Use FOIL to find that (x-6)(x+1) multiplies out to the equation as stated above. There is no fantastic way to get this other than to practice. Try the following and see if you get the right answers:

x^2 + 2x - 8 = 0; FACTORS (x + 4)(x - 2)
x^2 - 6x - 18 = 0; FACTORS (x - 6)(x + 3)
x^2 + x - 20 = 0; FACTORS (x + 5)(x - 4)
x^2 + 7x + 12 = 0; FACTORS (x + 4)(x + 3)

Note that you won't always have the coefficient of your squared term be 1, e.g. 4x^2 + 28 + 48. In this case, try dividing everything by the coefficient of the squared term (4), and proceeding.

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This won't always help either, because not all of your coefficients will be integers. In this case, i recommend using the quadratic formula:

[ -b (+ or -) sqrt(b^2 - 4ac) ] / [ 2a ]

Where the equation is of the form:

a(x ^ 2) + b(x) + c = 0.

Check the link below for more information:
http://en.wikipedia.org/wiki/Quadratic_formula#Quadratic_formula_or_Bhaskarach.C4.81rya.27s_Formula

NOTE: ALL THE EXAMPLES GIVEN ABOVE HAVE THE ANSWER BE ZERO. THIS MUST BE THE CASE WHEN FACTORING!!!

Good luck!!!

Answer 3

Always start with the equation =0.

Let's take something like 8x²-26x+21=0
First things first. The first term and last term are both positive, but the middle term is negative. Normally you won't mess around with the first term coefficient, so this means that the last term is the product of two negative numbers.

Now we need to do a prime factorization of the first term and the last term.
8 = 2*2*2
21 = 3*7
Somewhere in that mess, we need to find two products that, when added together, give us the middle term, 26. How about 4*3 and 2*7? 12 + 14 = 26.
So (4x-7)(2x-3) = 0. Note that the 4 and 3 need to be the outer coefficients to be multiplied together, and the 7 and 2 need to be the inner ones.. 4 and 2 need to be the first ones, and 7 and 3 need to be the last ones. There is only one way to make it work out.

Now, what if the last term is negative? Then it's the product of two negative numbers.
8x²-22x-21
Now we need to find the product of the first and last coefficients' factors that have a difference of the middle term (-22).
4*7 - 2*3 = 22. The sign is wrong, but we'll fix that when we put it into the factors.
(4x+3)(2x-7). There, now the 4 and 7 are multiplied and negative, and the 3 and 2 product will be positive, giving us -22.

got it? I hope so.

Answer 4

If you know how to FOIL--take a problem that looks like this:
(x+1)(x+2)
And turn it into this:
x^2+3x+2
Then you can factor. Factoring is simply FOILing in reverse. (In case you don't know, FOIL=First, Outer, Inner, Last--the steps you take to multiply 2 part x equations.)

So...we have...

x^2+4x+4

We know that to get x^2 (x squared) you have to multiply an x by an x...that gives us:

(x__)(x__) = x^2+4x+4

Our knowledge of foiling tells us that the last two terms in the ( ) must multiply to equal 4. (Because in FOIL, you multiply the last 2 terms.) We also know that the two terms in the ( ) have to ADD to equal 4. (Because in FOIL, you multiply the Inner and Outer portions, then add them.) So you have to think: What will multiply to give me 4 and add to give me 2?

The answer...2! So...

(x+2)(x+2)

Good luck with your math!

Answer 5

There's many ways to factor equations. I would give examples but Wikipedia explains it better.

Answer 6

you factor an equation by using some type of formula and then solving