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How to Factor a Quadratic Expression

Updated on December 6, 2012
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Quadratic Expressions

I love factoring. Most people do not have a love for all things math but I find factoring polynomials relaxing. Okay at least I have made you shake your head and smile, but let's find out how to factor.

Factoring in not the only method in finding the roots of a quadratic expression. Remember there are always two roots or answers to a quadratic. These problems can also be solved by graphing or using the quadratic formula.


What is Factoring?

We are going to find the roots by reworking the expression into two expressions in parenthesis. Remember not all quadratic equations can be solved in this manner.

So right now we should have an expression in this format ax2 + bx + c = 0

What we want is an equal expression that look like this (x+?)(x+?)

Keep in mind that is there is subtraction in the original problem that will change the operations in the parentheses.

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Example of Solving through Factoring

x2 + 11x + 30

Looking at this polynomial I know that both expression within the parentheses are going to have addition for their operations. I know because both operation in the polynomial are addition.

I also know that there are no coefficients of x.

(x+?)(x+?)

I need to think about two numbers that add to 11 and multiply to 30. Since 5 +6 =11 and 5x6=30 these are the two missing numbers.Since the operation are the same it does not matter which order they go in.

(x+5)(x+6)

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Factoring Example #2

The first example included only positive values.

Now we are going to try one with mixed operations.

x2-3x-88

I know that one expression is going to be addtion and one is going to be subtraction. Since we see the subtraction sign we know subtraction is involved but since two negative always equal a positive we can assume that since there is no addition in the last term that one of the expressions is addition

(x+?)(x-?)

There is still no coefficient of x.

So the first thing I am going to look at is what two numbers multiply to 88. The first number 8 & 11 multiply to 88 and they subtract from each other to make 3. So now I need to decide which is positive and which will the negative number. Since the 3 is a negative number the larger of the two numbers, 11, need to be negative.

(x+8)(x-11)


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