Which of the following equations has roots of zero and negative two? Circle the correct answer. The options are two 𝑥 multiplied by 𝑥 plus two equals zero, 𝑥 plus two all squared equals zero, 𝑥 minus two multiplied by 𝑥 plus two equals zero, or two 𝑥 multiplied by 𝑥 minus two is equal to zero.
So to work this out, what we need to do is find the values of 𝑥 that will make the left-hand side of the equation equal to zero. So I’m gonna go through each of our answers in turn to see if they have the roots of zero and negative two.
Well, what I’m gonna do is show you how you calculate the roots of each equation. But we could also check it out by putting in the values zero and negative two. So if we take a look at the first equation, which is two 𝑥 multiplied by 𝑥 plus two equals zero, we can see straight away that the first possible value or root of 𝑥 is gonna be equal to zero. And that’s because if we had two multiplied by zero, that’d be zero. And then zero multiplied by anything else is gonna give us a result of zero.
And then to find the next root, we need to see what 𝑥 would need to be to make the bracket equal to zero. And we can do that by making the bracket equal to zero. So we’ve got 𝑥 plus two is equal to zero. So therefore, what we need to do is subtract two from each side of the equation to see what 𝑥 would be equal to. So we know that 𝑥 is equal to negative two. So therefore, we’d say the roots of our equation will be zero and negative two. And these match the ones that we’re looking for.
So now what we need to do is just double-check the other equations. But this does look like it’s going to be our answer. And if we wanted to check, we could put the values back in. So if we put 𝑥 equals zero into the equation, we get zero multiplied by two. And that’s because we’d have a zero plus two. Well, zero multiplied by two is just equal to zero. And if we put negative two into the equation, we get two multiplied by negative two, which will give us negative four. And then this will be multiplied by zero. And that’s because negative two plus two is zero. So that would give us zero. So that’s correct.
So now what we’re gonna do is just check the other possible equations to see what their roots are. So if we look at the top right equation, well the top right would give us roots of two and negative two. And that’s because, in that equation, if we put two into the first bracket instead of 𝑥, we get two minus two, which give us zero. Zero multiplied by anything is zero. And in the second bracket, if we put negative two, we’d have negative two plus two, which again would give zero. Zero multiplied by anything is zero.
So now we can move down to the bottom left equation. Well, for the bottom left equation, we’d have repeated roots. And so the root will be negative two. And that’s because we’ve got 𝑥 plus two squared. If you put negative two in there, you’d have negative two plus two, which would again be zero. So zero is what we’re looking for as a result of that equation.
And then, finally, the bottom right equation, the root to this will be zero and two. And that’s because it’s very similar to the first one, except we’ve got 𝑥 minus two in the bracket. So therefore, we want two because two minus two is zero to give us our result of zero. So therefore, we can say that the equation with the roots zero and negative two is two 𝑥 multiplied by 𝑥 plus two is equal to zero.