Expert Mathmatician
Special Products
Perfect Squares:
Perfect squares are any numbers that can be squared into a whole number. However in Quadratics we can use Perfect Squares to our advantage. We can have a Binomial that is being squared, and convert it into a Trinomial. You might be confused as to how we can do this. Well, lets take an example (x+4)^2. Now to convert this Binomial into a Trinomial, we will write the equation as Binomial *Binomial { (x+4)*(x+4) }. Next, use F.OI.L and you will end up with a Quartic polynomial ( x^2+4x+4x+16 ). Finally collect like terms and you will have a Trinomial. You might be thinking "I dont wan't to do these steps, how can I make it easier?"
Well, a simpler rule would be to square the first term (in this case "x"), then add 2*the product of the first and second term (2*(4*x)), and finally add the second term squared (4^2). You will end up with the same answer but in a faster way. This rule applies in the same way if the two terms are being subtracted, but one step would change. Instead of adding 2*the product of the first and second term as our second step, we will subtract 2*the product of the first and second term.
Difference of Squares:
Difference of Squares are when you have two Binomials and you want to simplify it. You will always end up with a single binomial at the end. For starters, to use Difference of Squares, you need to have two binomials with one of them being added and the second needs to be subtracted (x+4)(x-4). Now you will use F.O.I.L and will end up with something like this (x^2+4x-4x+16). After collecting like terms you will have a Binomial (x^2-16) (note: for difference of squares you will always end up with the two numbers being subtracted). You end up with a Binomial because the two middle terms cancel each other out. Again, you might be saying "is there anyway to make this faster?"
Lucky for you, you can just square the first number and subtract that by the square of the second diget which would be { (x^2)-(4^2) }.
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