Find the solution set of the equation 𝑥 squared plus two 𝑥 equals zero in ℝ.
First, we are just gonna quickly draw our attention to the “in ℝ” part. What this just means is that we want the set of solutions that are in all real numbers. Okay, so now, we know where our solutions are gonna lie. All we need to do is actually solve our equation. On inspection of our equation, we can see that both terms actually contain 𝑥.
So therefore, to solve the equation, what we’re going to do is actually we’re going to factor our expression. And if we factor our expression, we’re gonna get 𝑥 outside the parentheses. And we get that because as we said, there is an 𝑥 in each term. And then, our first term inside the parentheses is gonna be 𝑥 because 𝑥 multiplied by 𝑥 gives us 𝑥 squared. And then, our second term is going to be positive two. And that’s because 𝑥 multiplied by positive two gives us positive two 𝑥.
Okay, great, we’ve now factored our equation. Now, let’s solve. In order to solve our equation, what we want to do is actually make 𝑥 equal to zero and 𝑥 plus two equal to zero because if the answer to our equation is zero, then either part — either the 𝑥 or the 𝑥 plus two — must also equal zero. So we found our first solution because it’s straightforward. It’s 𝑥 is equal to zero. And let’s find the second possible solution.
Well, if we have 𝑥 plus two is equal to zero, then if we subtract two from each side. We’re gonna get the solution 𝑥 is equal to negative two. So therefore, we can say that 𝑥 is equal to zero or negative two, which are both real numbers. So they satisfy the parameters for our answers or we can say that the solution set of the equation 𝑥 squared plus two 𝑥 is equal to zero is negative two, zero.