Find, in its simplest form, the quadratic equation whose roots are negative three and negative eight.
As it tells us that it’s a quadratic equation in the question, we actually know that we’re actually — if we were gonna factor it, we’d have a pair of parentheses. And also, as we know the roots of the equation, in that case, we can actually say that they’d be equal to zero. What we need to do now is actually to find out first what is inside our parentheses. And we know that for the equation to equal zero, one of the parentheses must also equal zero. So we can use that to actually find out what’s gonna go inside the parentheses. And we can do that because we know that the roots, so the 𝑥-values, are either negative three or negative eight.
So first of all, we can use negative three. And we know that negative three plus something is equal to zero. And we know that that’ll be negative three plus positive three. So therefore, we can actually include our first parentheses. So therefore, our first parentheses is 𝑥 plus three. Now we just need to find out what the second one could be. And now we need to see what will add to negative eight to give us zero, because that’s the other one of our roots. Well, the answer is positive eight. So therefore, great. We’ve now found what’s gonna go into our second parentheses, and that’s 𝑥 plus eight because we have positive eight from solving “what would add to negative eight to make zero”.
So great! We’ve now got the fact that, fully factored, our quadratic would be 𝑥 plus three, 𝑥 plus eight. And we can double check that because we can see that negative three plus three would be zero. That would make it totally equal to zero, the whole equation. And negative eight plus eight would also make the equation equal to zero.
Fantastic! Now we’d want to actually expand our parentheses. To do that, first of all, we’re gonna multiply the 𝑥 terms. So that’s 𝑥 multiplied by 𝑥 which gives us 𝑥 squared. And then I multiply 𝑥 by positive eight which gives us plus eight 𝑥. And then I multiply positive three by 𝑥 which gives us plus three 𝑥. And then finally, positive three multiplied by positive eight which gives us plus 24.
Fantastic! We’ve now expanded our parentheses and we’re actually almost there to the final answer. But if we look at the question, it says find, in its simplest form, the quadratic equation. So now what we need to do is collect like terms. So all we can see is that’ll be equal to 𝑥 squared and then plus. And we’ve got two like terms here. We’ve got eight 𝑥 and three 𝑥. So positive eight 𝑥 plus three 𝑥 which can give us positive 11𝑥 or plus 11𝑥. And then, just plus 24.
So therefore, we can say that, in its simplest form, the quadratic equation whose roots are negative three and negative eight is going to be equal to 𝑥 squared plus 11𝑥 plus 24.