Find the solution sets of the equation 18𝑥 squared plus 18𝑥 minus 36 equals zero in our set of real numbers.
On first inspection of our equation, we can actually see that it’s a quadratic cause we’ve got our 𝑥 squared term in there. But we can also see that actually each term has a common factor. The common factor right throughout is 18. So therefore, what I am actually gonna do is divide both sides of the equation by 18, which will give me 𝑥 squared, because 18𝑥 squared divided by 18 is 𝑥 squared, plus 𝑥, because 18𝑥 divided by 18 is 𝑥, and then minus two, because 36 divided by 18 is two. And this is all equal to zero because zero divided by 18 is still zero.
Okay, great. And as you can see, this is now in a much more manageable form. And we can actually go on and solve the equation. In order to solve this equation, what we’re actually gonna do is factor. So we’re gonna factor our quadratic. So we know that we’re gonna have two pairs of parentheses when we factor. I’m gonna have an 𝑥 at the beginning of each of them. And that’s because we have 𝑥 multiplied by 𝑥 gives us our 𝑥 squared term. And then, we’re gonna have a positive and a negative, one in each one of our parentheses. And we have that because we need to multiply two numbers together to give us negative two because it’s negative two. Then we obviously need to multiply a positive and a negative. It doesn’t matter which way round these go in our parentheses. It’s just I put them this way for now. And then within our parentheses, we’re gonna have two and one.
And we’re gonna have 𝑥 plus two and 𝑥 minus one. And we’ve got these as our factors because negative one multiplied by two is equal to negative two. So it’s great because that’s what we need to have as the answer to the product of our two factors. And also because negative one plus two is equal to one. And that’s what we want it to be because it’s actually the coefficient of our 𝑥 term. Obviously, usually there wouldn’t be the one written in there. But I’ve shown it just so we know what we’re looking for. And that’s positive one.
Okay, great. So we’ve now got our factors. So now we have that 𝑥 plus two multiplied by 𝑥 minus one is equal to zero. What we actually want to do to solve it, is we actually want to find out what are the possible values of 𝑥 that would allow our equation to be equal to zero. And to do that, we’re gonna make each of our parentheses equal to zero. Because if one of our parentheses is zero, then the final answer will also be zero.
So we’ve got 𝑥 plus two equals zero and 𝑥 minus one equals zero. So to find our first solution, we’re gonna solve 𝑥 plus two equals zero. To do that, we just subtract two from each side. And we get our first solution which is 𝑥 is equal to negative two. To get our second solution, we’re gonna solve 𝑥 minus one equals zero. And to do that, we add one to each side which gives us our second solution: 𝑥 is equal to one.
So therefore, we can say that the solution set of the equation 18𝑥 squared plus 18𝑥 minus 36 is equal to zero is: negative two, one.