### Video Transcript

Given that negative one and negative six are the solutions of the equation ๐ฅ squared plus ๐๐ฅ plus ๐ equals zero, find the values of ๐ and ๐.

So the first thing we can notice from our question is that weโre looking at a quadratic equation. And thatโs because itโs in the form ๐๐ฅ squared plus ๐๐ฅ plus ๐ equals zero. And what weโre also told our two solutions or two roots, and they are negative one and negative six. So therefore, from this information, what we can do is write our quadratic equation in factored form. And itโs gonna be ๐ฅ plus one multiplied by ๐ฅ plus six equals zero. But you might think, well, how do we know this is the factored form? Well, if weโve got a quadratic equation and itโs written in factored form, then the solutions are the values of ๐ฅ that will make each of our parentheses equal to zero.

So therefore, if we look at the right-hand parentheses, what we can see is if ๐ฅ was equal to negative six, weโd have negative six plus six, and that is equal to zero. And similarly, on the left-hand side, if we had the value of ๐ฅ that was negative one, then what weโd have is negative one plus one, which would be equal to zero. Okay, great! So weโve now got it in factored form. What do we need to do if we want to work out what our equation is in the form ๐ฅ squared plus ๐๐ฅ plus ๐ equals zero? I did mention earlier that it was ๐๐ฅ squared plus ๐๐ฅ plus ๐ equals zero is the general form. And weโre looking at a quadratic equation in this way.

However, we donโt need to worry about the ๐ because we already know in this question that thatโs just one because itโs a single ๐ฅ squared. Well, what weโre gonna do now is weโre gonna distribute across our parentheses to find out what our quadratic equation is. And in order to do that, what we need to do is multiply each term in the left-hand parentheses by each term in the right-hand parentheses. So first of all, weโve got ๐ฅ multiplied by ๐ฅ, which is ๐ฅ squared. Then we have this ๐ฅ multiplied by positive six, which is gonna give us positive six ๐ฅ. So weโve now got ๐ฅ squared plus six ๐ฅ. So thatโs the ๐ฅ multiplied by both the terms in the right-hand parentheses.

Now weโre gonna look at the positive one in the left-hand parentheses. Then weโve got positive one multiplied by ๐ฅ, which is gonna give us plus ๐ฅ. And then finally, weโve got plus six, and thatโs cause we had one multiplied by six. And then this is all equal to zero. Okay, great! Have we finished here? Well, no because thereโs one final thing we can do to simplify, and thatโs to collect our like terms. We can see here that weโve got six ๐ฅ plus ๐ฅ. So now, if we collect our like terms, what weโve got is ๐ฅ squared plus seven ๐ฅ plus six equals zero.

So have we finished here? Well, no because what we need to do is identify what the values of ๐ and ๐ are. So what we have is that ๐ is equal to seven and ๐ is equal to six, making sure that weโre careful here with the signs, but theyโre both positive, so this is okay. So we can say that given that negative one and negative six are the solutions to the equation ๐ฅ squared plus ๐๐ฅ plus ๐ equals zero, then the values of our ๐ and ๐, as we said, are seven and six, respectively.