Converting Standard Form to Factored Form

Quadratic equations in the standard form can often be converted to the factored form by factorization. To factorize an equation, you need a pair of numbers that multiply together to give the $c$ value of your quadratic expression, but add together to give you the $b$ value of your quadratic expression. Start with a standard form of quadratic expression.

$$y=x^2+7x+12$$

First of all, you need to make a list of the factors of 12, which is the c value of your quadratic expression (Table 1). Remember that you need to consider that numbers can have negative factors as well as positive factors, especially if you have to find factors of a negative number, which will have opposite signed numbers.

Table 1: Integers which multiply together to make 12 (the $c$ term of the standard form of quadratic expression, and all the values they add together to make (in order to identify a math with the $b$ term of the standard form of the quadratic expression. It is important to also consider that negative numbers can multiply together to get positive numbers, and negative numbers can multiply with positive numbers to make negative numbers. This is important to consider as the sum of the numbers may be a negative number. In this case, the pair of numbers which multiply together to get the $c$ value of 12 and add together to get the $b$ value of 7 are 3 and 4.
Number 1	Number 2	Product	Sum
1	12	12	13
2	6	12	8
3	4	12	7
-1	-12	12	-13
-2	-6	12	-8
-3	-4	12	-7

Also on this table we have the sum of each of these factors. This is because the sum of your factors needs to be equal to the $b$ value of the expression. We can see from this table that the factors of 3 and 4 have a product equal to the $c$ value of the quadratic expression (12), and a sum equal to the $b$ value of the expression (7). This gives us the factored form of the quadratic expression.

$$y=(x+4)(x+3)$$

Not every quadratic expression can be solved using this easy method. Sometimes quadratic expressions do not have a pair of integers that can be simply multiplied and added together to give the b and c values in the standard form. If there are not appropriate integers, it may be necessary to use the quadratic formula instead, or for more complicated expressions where there are a values that are not 1, you may need to find the greatest common factor, or use decomposition. Some quadratic expressions cannot be expressed in a factored form at all, as in these cases the curve does not cross the x axis, and so no factored form of the expression exists.