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One part stays the same, and that is the fact that we need to find a pair of numbers that will add up to the middle coefficient - in this case, -5. Instead of our two numbers needing to have a product equal to the constant on the end, we now need the product of our pair of numbers to be equal to the constant on the end times the leading coefficient.
It will be the same general idea, but there are a few extra steps to learn. This lesson will be on advanced factoring techniques, so I'm going to assume that you know a few things: first, that factoring is the process of breaking up a number into the things that we can multiply together to get that number.
This means that factoring a quadratic expression is the process of taking a trinomial and turning it into multiplication of two binomials - basically FOIL backwards. Well, the method I just described only works for quadratic trinomials where the - 5x - 3, we're going to be in trouble.
You can always quickly multiply out your answer to make sure you got the right thing, and if you do that here, looks like we're good. Not only is this example another one with a non-1 leading coefficient, it's also an example of a special quadratic that often messes students up. Now that we have these two values, we can fill in our area model.
We put the 9x and the -4 in one diagonal and the 6x and -6x in the other, but it actually doesn't matter which diagonal or even which order you put them in; they're all going to work!
Instead of simply being able to say -6 and 1 go in our binomials and we're done, there is an extra step before we can be sure of our answer.
While there are multiple ways of doing this step, I recommend using the area method to work your way backward to the answer.
Going down a row to the bottom, 1x and -3, they don't have any factors with the numbers in common, and they also don't have any variables in common, which means the only thing I can divide out is a 1. What we now have on the left and above our little area model is our factored answer. For this one, they'll have to add up to zero (the middle term) and multiply to -36, which was the 9 in front times the -4 on the end.
The terms that are on the same side are the terms that go in parentheses together to make up our two binomials, and I end up with (2x 1)(x - 3). Quickly looking through our options here and knowing that we're going to have to add up to zero makes it pretty obvious that 6 and -6 are going to be our winners.
To be in the correct form, you must remove all parentheses from each side of the equation by distributing, combine all like terms, and finally set the equation equal to zero with the terms written in descending order. In this case, we need to remove all parentheses by distributing, combine like terms, and set the equation equal to zero with the terms written in descending order. In this case, we need to remove all parentheses by distributing, combine like terms, and set the equation equal to zero with the terms written in descending order.
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