Given that negative one and negative six are the solutions of the equation 𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, find the values of 𝑏 and 𝑐.
So the first thing we can notice from our question is that we’re looking at a quadratic equation. And that’s because it’s in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. And what we’re 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 it’s gonna be 𝑥 plus one multiplied by 𝑥 plus six equals zero. But you might think, well, how do we know this is the factored form? Well, if we’ve got a quadratic equation and it’s written in factored form, then the solutions are the values of 𝑥 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 𝑥 was equal to negative six, we’d have negative six plus six, and that is equal to zero. And similarly, on the left-hand side, if we had the value of 𝑥 that was negative one, then what we’d have is negative one plus one, which would be equal to zero. Okay, great! So we’ve 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 𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero? I did mention earlier that it was 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero is the general form. And we’re looking at a quadratic equation in this way.
However, we don’t need to worry about the 𝑎 because we already know in this question that that’s just one because it’s a single 𝑥 squared. Well, what we’re gonna do now is we’re 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, we’ve got 𝑥 multiplied by 𝑥, which is 𝑥 squared. Then we have this 𝑥 multiplied by positive six, which is gonna give us positive six 𝑥. So we’ve now got 𝑥 squared plus six 𝑥. So that’s the 𝑥 multiplied by both the terms in the right-hand parentheses.
Now we’re gonna look at the positive one in the left-hand parentheses. Then we’ve got positive one multiplied by 𝑥, which is gonna give us plus 𝑥. And then finally, we’ve got plus six, and that’s cause we had one multiplied by six. And then this is all equal to zero. Okay, great! Have we finished here? Well, no because there’s one final thing we can do to simplify, and that’s to collect our like terms. We can see here that we’ve got six 𝑥 plus 𝑥. So now, if we collect our like terms, what we’ve got is 𝑥 squared plus seven 𝑥 plus six equals zero.
So have we finished here? Well, no because what we need to do is identify what the values of 𝑏 and 𝑐 are. So what we have is that 𝑏 is equal to seven and 𝑐 is equal to six, making sure that we’re careful here with the signs, but they’re both positive, so this is okay. So we can say that given that negative one and negative six are the solutions to the equation 𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, then the values of our 𝑏 and 𝑐, as we said, are seven and six, respectively.