Video: Solving by Taking Roots

Learn how to rearrange equations, such as 2(𝑥 + 4)² = 162, and take square roots in order to find the possible missing values of the unknown variable.


Video Transcript

Solve by taking roots

So in this question, we must solve for 𝑥, finding 𝑥 when we’ve got three 𝑥 squared minus nineteen equals two hundred and eighty-one. So what we want to do essentially is to get 𝑥 on its own. So we’re going to do the opposite of the order of operations. So first of all, we can see we can nicely and easily just get away that subtraction. And the way to get rid of a subtraction is you add. So we’re going to have to add nineteen to both sides. Remember whatever you do to one side, you must do to the other.

Adding nineteen to both sides will give us three 𝑥 squared on the left-hand side. And that would be equal to two hundred and eighty-one plus nineteen which is three hundred. And then we can see that now on the 𝑥 squared we’ve got three multiplied by 𝑥 squared. Well the opposite of times by is divide by. So getting rid of that first, we’re going to have to divide both sides by three.

As you must remember, whatever you do to one side we do it to the other. So now we’ve got 𝑥 squared is equal to one hundred. And this is the only bit that’s any different. So the bit that we need to be careful with is the opposite of square is square root. So here you can see it’s 𝑥 squared. So to get rid of the squared, we must square root both sides, giving us 𝑥 equals, and we might be tempted to write just ten. But that’s not strictly true, because we know as we’ve just done, we square rooted both sides. We know that ten times ten is equal to a hundred. That so is negative ten multiplied by negative ten. So when we square root, when we take roots, we have to put the answer as plus or minus, in this case ten. And we write it a little bit like that. So there we have it. We have solved this quadratic by taking roots. We just simply rearranged to make 𝑥 the subject, well 𝑥 squared the subject. And then we square rooted to get 𝑥 the subject.

Let’s have a look at another quadratic that’s slightly different.

Solve two multiplied by all of 𝑥 plus four all squared equals one hundred and sixty two. So we can see the only thing we can get rid of away from the 𝑥 side is this two multiplied by. We know the opposite of times by is divide by, so now we must divide both sides by two. This will give us 𝑥 plus four all squared on the left-hand side and eighty one on the right. And now we have two options here. We can either take roots, which in this case we’re going to do cause that’s what the video is about. Or we could know that 𝑥 plus four all squared is the same as. And we could multiply out those brackets using foil. And then we could get a quadratic that we would be able to solve. Or if we couldn’t factor it, then maybe we would use a quadratic formula. But in this case it’s a lot quicker if we just take roots.

So doing that on the left-hand side, we get 𝑥 plus four. But the right-hand side, we end up with two options. We end up with nine or negative nine. Because we’ve just shown in the example before if we’ve got negative nine and we multiplied it by negative nine, we get positive eighty-one. And then if we have nine And multiplied it by nine, we also get positive eighty-one. So square rooting the eighty-one gives us plus or minus nine.

And now we have two options for a root. We have an option where the nine is positive and we have one for where it’s negative. So in the case where it is positive, we’ve got nine and then we take away four. And we know that nine take away four is just five. And in the case where it’s negative, we’ve got negative nine take away four. And negative nine take away four is equal to negative thirteen. So there we have it. We have solved by taking roots for this quadratic. And it’s a lot simpler than as we said multiplying out the brackets and then putting it into a quadratic and finding out if we can factor it or not. In this case, it’s a lot easier just to take roots. But the most important thing is remembering that plus or minus when we square root.

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