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More Challenging Quadratic Factoring

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Today we're going to continue our conversation about factoring quadratics. And we're going to look at some more challenging examples where the coefficient in front of the x squared term is not 1. So we're going to start by reviewing how to factor if the x squared coefficient is 1. And then we'll do some more of the challenging examples.

So for my first example, I'm going to factor the expression x squared minus 2x minus 15. I know that I'm looking for two integers that when added together will give me the value of my coefficient of my x term-- so negative 2-- and when multiplied together will give me the value of my constant term, or negative 15. So I'm going to start by making a list of pairs of integers that multiply to give me negative 15.

So that would be positive 1 and negative 15, negative 1 and positive 15, negative 3 and positive 5, or positive 3 and negative 5. So now I need to see which of these pairs of numbers when the two numbers are added together will give me a value of negative 2. Positive 1 and negative 15 added will give me negative 14. Negative 1 and positive 15 added will give me positive 14. Negative 3 and positive 5 added will give me a positive 2. And 3 and negative 5 finally gives me the value I'm looking for of negative 2.

So I know that my two integers I'm looking for are positive 3 and negative 5. And I can use those to write the factored form of this expression. I'll have x plus 3 and then multiply by x minus 5. So this is the factored form of this expression, which was in expanded form.

So let's look at another example of writing an expression in factored form. This example is a little bit different, because my x squared term has a coefficient that is different from 1. So we're going to use the process of factoring in a little bit different way.

So I know that when I'm factoring, I want to write the expression to look something like this-- some value m times x plus an integer p multiplied by some value of n times x plus the integer q. And if I were to think about the process of FOILing, I know that in my factored form, the product of my first two terms is going to give me my first term in the expanded form. In other words, my values for m and n when multiplied together should equal the coefficient of my x squared term-- so 2.

And the only two numbers that multiply to give me 2 are 2 and 1. So I know that in factored form, my expression will look something like this-- 2 times x plus an integer p multiplied by 1 times x, or just x, plus an integer q. So now when I multiply 2x times x, that does give me 2x squared.

So now I need to find my integers p and q so that, again, when I multiply p and q together, they will give me a negative 8. But now when I am looking for what will give me my coefficient of negative 6, I have to think about the fact that I'm going to be multiplying my 2x term by my value of q. So now I'm looking for integers p and q such that 2 times q plus p will give me and my coefficient of my x term-- so negative 6.

So let's start by making a list of what multiplies to give me negative 8. The pairs of numbers that will multiply to give me negative 8 include negative 4 and positive 2, positive 4 and negative 2, negative 8 and positive 1, and positive 8 and negative 1. So I'm looking for which of these values will satisfy this for p and q.

So looking at my first possibility, if q is negative 4, 2 times negative 4 gives me negative 8. And substituting 2 in for p, negative 8 plus 2 will give me negative 6. So I found that my value for q should be negative 4 and my value of p should be positive 2.

So going back to my factored expression, I will have 2x plus 2 times x minus 4. So this is the factored form of my original expression. We can verify that by using FOIL to multiply these two together. And that should equal our expanded form.

So 2x plus 2 times x minus 4-- my first two terms multiplied together will give me 2x squared. My outside terms multiplied together will give me negative 8x. My inside terms multiplied will give me positive 2x. And my last two terms multiplied will give me negative 8. I can combine these two like terms, which will give me negative 6x, bring down my other two terms, and I have 2x squared minus 6x minus 8, which matches my original expression.

So for our last example, I'm going to show you another method of factoring a quadratic expression. So I'm going to start by noticing my coefficient of my x squared term and my constant term and multiplying them together. So 4 times negative 2 will give me negative 8.

Then I'm going to list out the factors of negative 8. So that would include 4 and negative 2, negative 4 and positive 2, 8 and negative 1, and negative 8 and positive 1. I'm looking for which pair will also have a sum of my coefficient of my x term-- so 7. And I see that the pair of numbers that has a sum of 7 is positive 8 and negative 1.

So my next step is to create a 2 by 2 table. In this table, I'm always going to put my term from my original expression, my x squared term, in the upper left hand corner-- so 4x squared. And I'm also always going to put my constant term in the lower right hand corner-- so negative 2.

Then I'm going to take the values that I found from my list and make those coefficients of my 2x terms. And I can put those in either of these two boxes. So I'll put positive 8x here and negative 1x here.

My next step is to factor out a term from each of the terms in my first row and in my second row. So the greatest common factor of 4x squared and 8x is going to give me 4x. And my greatest common factor of negative 1x and negative 2 is negative 1.

I'll do the same thing, and factor out something from the terms in each column. So the greatest common factor of 4x squared and negative 1x is 1x. And the greatest common factor of 8x and negative 2 is 2.

So now these two quantities will represent the terms in my first factor of this expression. And these quantities will represent the terms of my second factor of this expression. So in factored form, this will look like 4x minus 1 and multiplied by 1x plus 2.

So I can verify that this is in fact the factored form of this original expression by using FOIL to multiply these two binomials. So 4x minus 1 multiplied by 1x, or just x, plus 2-- my first two terms multiplied together will give me 4x squared. My outside terms multiplied will give me 8x. My inside terms will give me negative 1x. And my outside will give me a negative 2.

I can combine my two middle terms. Positive 8x minus 1x will give me 7x, which is positive. Bringing down my other two terms, 4x squared plus 7x minus 2 matches my original expression. So my answer, which is in factored form, is correct.

So let's go over our key points from today. Factoring is the process of writing an equation from expanded form to factored form. Factoring involves finding two integers whose sum is the coefficient of the x term and whose product is the constant term of quadratic equations with an x squared term coefficient of 1. And the product of the coefficients of the x terms in factored form must equal the coefficient of x squared term in the expanded form.

So I hope that these key points and examples helped you understand a little bit more about factoring more challenging quadratic expressions. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.