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Factor quadratics with an 𝑥 Squared Coefficient of One
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Factoring is the conversion of an expression to an equivalent form with one term, and it is the opposite of the distributive property.
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So as a reminder, the distributive property is that 𝑎 multiplied by all of 𝑏 plus 𝑐 is equal to 𝑎𝑏 plus 𝑎𝑐.
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So the opposite of that is factoring, so 𝑎𝑏 plus 𝑎𝑐 is equal to 𝑎 multiplied by all of 𝑏 plus 𝑐, where 𝑎 is the greatest common factor or GCF for short.
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So this takes us from two terms to one term which is fully factored now let’s have a go at actually factoring a quadratic.
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So if we have to factor the quadratic 𝑥 squared plus seven 𝑥 plus twelve.
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Then we need to remember first is that every quadratic comes in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 where 𝑎, 𝑏, and 𝑐 are all constants.
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So in this case, we can see that our 𝑎, our coefficient, of the 𝑥 squared is equal to one.
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So we don’t need to worry about that too much.
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But to factor a quadratic, the first thing we need to do is find its factors.
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And to do that, we need to find what they add to get and what they multiply to get.
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In another words, what’s their sum and what’s their product?
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So we need a table.
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And their sum will always be 𝑏 whereas their product will be 𝑎 multiplied by 𝑐.
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But as I said a moment ago, we don’t need to worry about it too much in this case because it’s equal to one.
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So we can see for this one if we draw a table, its sum would be 𝑏, which is seven, and 𝑐 being twelve.
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So we need to now look through the factor pairs of twelve and see which of those could add to get seven.
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So first of all, one and twelve, nope that doesn’t add to get seven.
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So how about two and six?
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Nope And then finally, three and four.
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So yea, three add four is seven, and I’m sure that many of you knew that that factor pair was what we we’re looking for before you even had to do anything.
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But it’s always good to check through.
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So because 𝑎 is equal to one, we can just write the brackets with the 𝑥s in there straight away.
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And we say 𝑥 add three all multiplied by 𝑥 add four.
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So this example, we had only positive numbers.
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For our next two examples, we’re gonna look at how it’s different when we have negative numbers.
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So we have to factor 𝑥 squared minus four 𝑥 plus three.
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We know the very first thing we always have to do is say what’s the product and what’s the sum.
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So what’s it add to get?
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Well it adds to get negative four, and it multiplies to get three.
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So we have two numbers that multiply to give a positive number that add to give a negative, so we know that a negative multiplied by a negative is equal to a positive.
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So therefore, because they add to give a negative and multiply to give a positive, then we’re gonna be looking for two negative numbers.
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But anyway, let’s list the only factor pair of three, because it’s prime, and that is one and three.
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Well if both of those are negative, they add to give negative four.
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So it works, so we’re gonna write our brackets out straight away, our parenthesis, and put 𝑥 in them and then our factors.
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So 𝑥 minus one all multiplied by 𝑥 minus three.
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There we have it.
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Done!
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Okay so there’s one more type of negative numbers when you’re factoring so let’s look at that so we need to factor 𝑥 squared minus 𝑥 minus thirty.
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Again we need a table.
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We say what’s it add to get what’s it times to get.
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So having a look at the coefficient in front of the 𝑥, kinda looks that there isn’t one.
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We have to remember that that is hiding a one.
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So that was negative one 𝑥, so in this case it adds to get our factors; add to get minus one, and then multiply to get negative thirty.
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So as we have two factors that multiply to give a negative number, that means that one of them must be positive and one of them must be negative, as a negative multiplied by a positive is equal to a negative.
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So now listing the factor pairs of thirty, we’ve got one and thirty.
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Now we’re looking for a difference of one; one and thirty do not have a difference of one so it won’t be them.
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And then two goes in fifteen times, and they don’t have a difference of one.
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Three goes in ten times.
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Again, no difference of one.
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Four does not go in.
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But five does, and five goes in six times.
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And happily, they have a difference of one.
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But now we need to work out which one is gonna be negative.
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So because when they add together they give us a negative number, that means the larger number, or in this case six, has to be negative.
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So we’re gonna put our parentheses straight down, put the 𝑥s straight in them and then pop in the factors.
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So we have 𝑥 plus five all multiplied by 𝑥 minus six.
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And there we have it.
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Done!
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So we must remember when we are factoring quadratics, the very first thing we need to do is do a table.
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So what’s add to get what’s it times to get?
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And then once we’ve done that, we need to focus on what are our negatives and positives.
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So we know negative multiplied by a positive is equal to a negative, and so on and so for.
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We list the factor of pairs, finally work out which ones are going to be negative if they are, and then you put them straight into parentheses.
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So we have learned how to factor when the coefficient of 𝑥 squared is equal to one.