### Video Transcript

Determine whether the following expression is prime or not: two ๐ฅ squared minus seven ๐ฅ minus four. If it is not prime, give the expression in its factorized form.

Well, if an expression is prime, what it means is that it cannot be factorized any further. So letโs work out whether our expression is in fact a prime. Well, to help us decide whether our expression is prime or not, what we can use is something called the discriminant. Well, what the discriminant is something thatโs really useful. So if weโve got a quadratic in the form ๐๐ฅ squared plus ๐๐ฅ plus ๐, then the discriminant is ๐ squared minus four ๐๐. But howโs this gonna be useful for our problem? Well, what we know is that if the discriminant ๐ squared minus four ๐๐ is greater than zero, then there are two real roots.

So therefore, we know that our expression would not be prime. And we also note if ๐ squared minus four ๐๐ was equal to zero, then weโve got a repeated root. So again, our expression would not be prime. And what we also know is that if ๐ squared minus four ๐๐ is less than zero, then there are no real roots. And this is when we know that our expression is gonna be prime because it cannot be factorized. Great, so now we know what the discriminant is and we know how to use it. Letโs try using it with our expression. Well, ๐ is equal to two, ๐ is equal to negative seven, and ๐ is equal to negative four in our expression. So thatโs our ๐, ๐, and ๐ because our ๐ is the coefficient of ๐ฅ squared, ๐ is the coefficient of ๐ฅ, and then weโve got ๐, which is our numerical value on the end.

Well then, therefore, if we substitute our values in, weโre gonna get ๐ squared minus four ๐๐. So our discriminant is equal to negative seven squared minus four multiplied by two multiplied by negative four which is gonna be equal to 49 plus 32. And this could be four multiplied by two multiplied by negative four is negative 32. And if we subtract a negative, itโs the same as adding. So this is gonna be equal to 81. So therefore, we can say that definitely our expression is not prime. So we can factorize it. So now what we want to do is factor or factorize our quadratic. And to do that, what we need to do is put it into two pairs of parentheses. And we could do that with some trial and error. However, there is a method that we can use if you want to, which makes things easier when weโve got coefficient of ๐ฅ squared greater than one.

Well, first of all what you do is multiply ๐ and ๐. So in this case, thatโs gonna be two multiplied by negative four which is gonna give the answer negative eight. And then what we want to find is two factors whose product is negative eight which we just found and whose sum is the coefficient of ๐ฅ, which is negative seven. Well, our two factors are gonna be negative eight and positive one. Thatโs because negative eight multiplied by positive one is equal to negative eight and negative eight add one is equal to negative seven.

Okay, great, so now whatโs the next step? So now, what we do is we split up our coefficient of ๐ฅ into the two factors we got, which was negative eight and positive one. So weโre gonna have two ๐ฅ squared minus eight ๐ฅ plus one ๐ฅ minus four. You wouldnโt usually write the one ๐ฅ but Iโm putting the one in front of the ๐ฅ just cause itโs gonna help with the next stage. We can see whatโs happening. So now, what we need to do is factor or factorize the first two terms together and then the second two terms. Well, if we factorize the first two terms, weโre gonna take two ๐ฅ out as a factor. And then inside the parentheses, weโll have ๐ฅ minus four. And thatโs because two ๐ฅ multiplied by ๐ฅ is two ๐ฅ squared and two ๐ฅ multiplied by negative four is negative eight ๐ฅ. And then we take out one as a factor of our last two terms. So what weโre gonna have is plus one then multiplied by ๐ฅ minus four.

So now, to form our two factors, what we do is take the first part before both of the parentheses. So we got two ๐ฅ and then add one. And then for a second pair of parentheses what weโre going to take is the value inside the parentheses in this line, so ๐ฅ minus four. Itโs also a good check because we know that weโre on the right track because we have the exact same one in each of our parentheses ๐ฅ minus four. If these were different, then youโd have done something wrong. Then you need to check. So therefore, weโre gonna get two ๐ฅ plus one multiplied by ๐ฅ minus four.

So therefore, weโve solved the problem cause weโve shown that the expression is not prime. And weโve shown that its fully factorized form is two ๐ฅ plus one multiplied by ๐ฅ minus four. But what we can do now quickly is check this by distributing across the parentheses. So if you have two ๐ฅ multiplied by ๐ฅ, itโs two ๐ฅ squared. Two ๐ฅ multiplied by negative four gives us negative eight ๐ฅ. One multiplied by ๐ฅ gives us plus ๐ฅ. And then finally, positive one multiplied by negative four gives us negative four. So then if we simplify, we get two ๐ฅ squared minus seven ๐ฅ minus four which is what we started with. So weโve doubled checked and we definitely have the correct answer.