The first thing you must learn in my tutorial is how to factor quadratics, the simplest way to solve. Factoring is breaking up an equation or expression into two or more simpler parts that are being multiplied together. This can even be done with numbers. Factor 42. 3 * 7 * 2 are its factors. Now you will learn how to do this with variables.
First we will learn about a very simple type of factoring. Example: 3x + 6. All you have to do is "pull out" the three: 3(x + 2). Check it and you multiply 3 by x and 3 by two to get 3x + 6. Example: 2x³ + 4x² + 6x.
We can pull out 2x to give us 2x(x² + 2x + 3). Check it and it works! Sometimes, however, the first coefficient is negative while the rest are positive. In this case pull out a negative, even if its only -1, and change the others accordingly. Example: -4x² + x + 3 changes to -(4x² - x - 3).
Next are perfect squares. These are quadratics that not only can be factored into two expressions, but the expressions are always the same.
x² + 2x + 1 can be factored into (x + 1)(x + 1) or (x + 1)². To apply this method to any perfect square you must first know two things:
1) to FOIL - this is to multiply first, outer, inner, last which means to multiply two expressions into a quadratic by multiplying the first term of the first expression by the first term of the second expression, adding it to the product of the first term of the first expression and the second term of the second expression, adding this to the product of second term of the first expression and the first term of the second expression, finally adding this to the product of the second term of the first expression and the second term of the second expression.
Now that your head is filled with gobbledygook, please clear your mind and take into consideration the following diagrams.
2) y = ax² + bx + c is the general term for a quadratic. So if you have x² + 6x + 9 then a = 1, b = 6, and c = 9. There is no number in front of x² for a, so it is assumed to be 1. This phrasing of quadratics is especially helpful in more difficult topics and should be remembered.
Now you are ready to learn perfect squares, full fledged. With x² + 6x + 9, or any perfect square, look at the a term and the c term to see if they are all squares. 1 and 9 both check out. Then set up an expression like this (x + 3)(x + 3). We know from above that this is the right answer, but to make sure we need to see that the b term comes out as six, and therefore we should multiply again to check. The way we simplified the quadratic to (x + 3)² is that the first term is the square route of a and the latter term is the square route of c.
A square route of a number is the exact number that can be squared to give the original. For example: the square route of 4, or Ö(4), is 2 because 2 * 2 = 4 and 2 =2.
Another type of perfect square is where a (the x² coefficient) is not 1. For example: 4x² + 4x + 1
The a term is a square and c is a square so this is a perfect square quadratic. There is one more thing to check, though. FOIL the two expressions and find if the b term comes out right.
So the expression is (2x + 1)². Check it and it works!
Here is a game to practice on.
Now, most things in life aren't perfect and the same is true with quadratics. For a general quadratic, say x² + 5x + 4, you need to use trial and error. First take the factors of the c term, 4:
1 and 4
2 and 2
Then see which ones will add to give you the b term: 2 + 2 = 4, nope that does not work.
1 + 4 = 5, Yay! It works, so the equation factors into (x + 1)(x + 4). Even though you think this is right, check it anyway using FOIL. x * x = x², x * 4 = 4x, 1 * x = x, and 1 * 4 = 4.
Add them up, and you get: x² + 4x + x + 4
Which simplifies to x² + 5x + 4 which is correct!
The factors of c trick only works if the a term is 1.
Let's try another one. 2x² + 9x - 5.
Now we start by factoring the a term, 2:
2 and 1
1 and 2
Now we factor the c term, 5:
-1 and 5
5 and -1
-5 and 1
1 and -5
So we multiply, one at a time. (2 * -1) + (5 * 1) = 3 nope
(1 * -1) + (2 * 5) = 9: It Works!
Then we get (1x + 5)(2x - 1), check it and it works. Note that the numbers being multiplied were put in opposite brackets: the 2 and 5 were multiplied so they go in opposite brackets.
Another type of simple factoring is known as the difference between two squares. This works with a subtraction of squares like this: x² - 36. Take the square route of the first term, x, and set up an expression like this (x + ?)(x - ?). Replace the question mark with the square route of the second term, 6, and set the expression like this (x + 6)(x - 6). Check by FOILing and you'll see it works. Sometimes, however, you'll find a problem like this: x4 - 256. It factors into (x² + 16)(x² - 16), but wait! The second expression is also the difference between two squares so it can be factored like so: (x² + 16)(x + 4)(x - 4). Sometimes this can happen several times so be careful and be sure to factor completely.
In certain situations you might not be able to factor properly because you need to factor differently in the beginning. Example: 3x³ + 36x² + 108x. First factor like you first learned by pulling out a single term. Can you tell what term it is? 3x! If you pull it out you get 3x(x² + 12x + 36). Then you can factor the other part to (x + 6)(x + 6), a perfect square. You can leave it as 3x(x + 6)² and you're done.
Here are some practice problems to do. Use them well to prepare for the ultimate battle against Mr. Linear!
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