方程根方程
成绩单
Equations with square roots. The test sometimes gives us an equation to solve involving the square root.
In such an equation, the variable will appear under the radical. So for example, this would be an equation with square roots.
Square root of x plus 3 equals x minus 3. We'll solve this one later in the video.
在此，这是我们在本课程中谈论的等式的类型。当然，我们通过平方撤消平方根，和 我们总是被允许平方两侧。有时，对于 the simplest radical equations, all we have to do is square both sides. So, for example, if we have something like, square root of x plus 2 equals 3.
好吧，只需平方两侧，我们在左边得到x加2，我们在右边得到9，减去，我们得到x等于7。 Fantastic. But, that equation was a bit too simple to appear on the test. The test is not actually gonna hand us something simple on a silver platter like that, it's gonna be a little trickier.
Before we go on, let's think about this. Is it always true, for any value of k, that if we take the square root of k squared, that we'll get back to k? That is to say, that the square root undoes the squaring and 总是把我们送回我们从哪里开始。这总是真的吗？
And of course, the answer is no. The equation is true for positive numbers and for 0. 但不是为了k的负值。例如，如果K等于负4， then of course, when we square it we'll get positive 16. Negative 4 squared is positive 16.
And when we take a square root of 16, we get 4. In other words, we don't go back to the original starting number. So that's important. This suggests we may run into some kinds of problems when negative values arise. 所以这是我们的雷达。当我们获得负值时会发生什么， 特别是，当自由基下的东西是负面的？
我们必须注意这一点。事实证明，在激进的方程中，我们必须了解无关根。 当我们正确地完成所有代数时，包括平衡方程的两侧， 代数可以导致答案实际上在原始方程中实际上工作。这些是无关的根源。
So I want to emphasize, this is not about making a mistake, in other words, even if we do all the algebra correctly, just by virtue of the fact that we square, we produce extra roots, extraneous roots that are not ones that actually solve the original equation. 重要的是要理解，即使正确地完成所有代数，外来根也会在自由基方程中出现。
Now we can look at the, the equation we had at the beginning. So, here's the equation from the beginning. So of course, what we'll do, is we'll square both sides. Of course that, that binomial squared on the right side we, we, we foil that out to x squared minus 6x plus 9. You may remember the pattern for the square of a difference.
Then we'll gather everything on one side, so we get a quadratic equal to 0. We'll factor that, it's very easy to factor, and we get two roots, 1 and 6. Now normally with algebra, you'd think okay, we must be done, we found the value of X. 但是利用自由基方程，我们必须小心。我们是否知道这两个根源都在工作？
Maybe they both do, or maybe one of them is an extraneous root. So we have to check our answers. We have to check each answer we found to make sure they worked, because right, now just looking at them 1 and 6, we don't know. Are those both true roots, are they both extraneous roots? Do they work in the original equation?
The only way we find is by plugging them in. So, here's the original equation. Here are the roots that we found from the algebra. So first of all we're going to check the first one, x equals 1. 将其插入左侧，我们得到1加3平方根4的平方根，它为2.将其插入右侧，我们得到1减3，为负2。
So the two sides of the equation are not equal. One side equals 2, one side equals negative 2. 所以这个根不起作用。现在，我们将检查另一个。 Plug it in to the left side, we get square root of 6 plus 3, of course that's 9. Square root of 9 is 3.
On the other side we get, 6 minus 3, which is also 3. The two sides work, so that one does legitimately work. And it does solved the problem. So, this equation has one solution that works, x equals 6. 这是唯一有效的解决方案。x等于无关根，因为即使 我们跟着代数正确,即使the algebra gave us that root, that root does not actually work in the original equation.
We need to square both sides to undo the radical, but this very act can produce extraneous roots. If we get a quadratic after squaring, which is common on the test, the algebra will lead to two roots. 有时两个根都是工作。有时一个根作用，一个是无关紧要的。
Sometimes both are extraneous, and the equation has no solution. So, here's a practice problem. Pause the video, and then we'll talk about this. 好的。所以，在这里，我们在双方都有激进的。
Radical equals radical. So of course, we're just gonna square both sides. 我们得到2x减去2等于x减去4.嗯，非常简单的方程来解决。 And we get x equals negative 2. All right, very good.
但now, what happens here if we plug this back into the original equation? When we plug this in, this results in the square root of a negative on both sides. 因此，我们得到了负6的平方根，负6的平方根是实数系之外的东西，它不会在数字线上的任何地方。 所以我们不能用它来做数学。这只是为了我们的目的，这只是一个错误和 该等式没有解决方案。
Finally, keep in mind that we should square both sides only when the radical is by itself on one side of the equation. If the radical appears on, with other terms on one side, we will have to isolate the radical on one side, before it would make sense to square both sides. So, here's a practice problem video, and then we'll talk about this.
Okay, so we do not have the radical by itself. So the very first thing we have to do, is subtract that 2 from both sides. So we get the radical, 4 minus 3x equals x minus 2. 现在我们可以平方两侧。当然，我们得到了差异的平方。
The square of that binomial. And that expands out to x squared minus 4x plus 4. 现在,我们要两边同时减去4,dd 3X to both sides and this will lead us to x squared minus x. 非常容易因素，对X次x减去1.而代数导致我们的解决方案x等于0和x等于1。
现在我们需要检查这些答案。 好的。So those are the roots that the algebra found for us. First of all, check x equals 0. Plug this into the left side, and 我们得到的是2加根4，这是2加2，即4。
将其插入右侧，它是0.当然，4不等于0。 So this one does not work. So this would be an extraneous root. 现在检查x等于1.将其插入左侧，我们得到2加4.3倍1，所以4减3。
And of course, that would be 1. And so, that's gonna be 2 plus 1 which is 3. 当然，这不等于1.不等于x，在等式的另一侧是1。 So this one doesn't work either. And so, neither one of the roots that the algebra gave us works.
So this equation simply has no solution. Both the solutions that the algebra gave, were extraneous roots. In summary, to undo a radical equation, we need to square both sides. We have to mo, move something else to the other side sometimes, 在平方之前隔离激进术。换句话说，我们本身需要激进的。
So there are other terms on that side with the radical. We need to get rid of them, move them to the other side before we can square. And the very act of squaring produces extraneous roots, therefore, we must check each answer the algebra gives us back in the original equation.
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在此，这是我们在本课程中谈论的等式的类型。当然，我们通过平方撤消平方根，和 我们总是被允许平方两侧。有时，对于 the simplest radical equations, all we have to do is square both sides. So, for example, if we have something like, square root of x plus 2 equals 3.
好吧，只需平方两侧，我们在左边得到x加2，我们在右边得到9，减去，我们得到x等于7。 Fantastic. But, that equation was a bit too simple to appear on the test. The test is not actually gonna hand us something simple on a silver platter like that, it's gonna be a little trickier.
Before we go on, let's think about this. Is it always true, for any value of k, that if we take the square root of k squared, that we'll get back to k? That is to say, that the square root undoes the squaring and 总是把我们送回我们从哪里开始。这总是真的吗？
And of course, the answer is no. The equation is true for positive numbers and for 0. 但不是为了k的负值。例如，如果K等于负4， then of course, when we square it we'll get positive 16. Negative 4 squared is positive 16.
And when we take a square root of 16, we get 4. In other words, we don't go back to the original starting number. So that's important. This suggests we may run into some kinds of problems when negative values arise. 所以这是我们的雷达。当我们获得负值时会发生什么， 特别是，当自由基下的东西是负面的？
我们必须注意这一点。事实证明，在激进的方程中，我们必须了解无关根。 当我们正确地完成所有代数时，包括平衡方程的两侧， 代数可以导致答案实际上在原始方程中实际上工作。这些是无关的根源。
So I want to emphasize, this is not about making a mistake, in other words, even if we do all the algebra correctly, just by virtue of the fact that we square, we produce extra roots, extraneous roots that are not ones that actually solve the original equation. 重要的是要理解，即使正确地完成所有代数，外来根也会在自由基方程中出现。
Now we can look at the, the equation we had at the beginning. So, here's the equation from the beginning. So of course, what we'll do, is we'll square both sides. Of course that, that binomial squared on the right side we, we, we foil that out to x squared minus 6x plus 9. You may remember the pattern for the square of a difference.
Then we'll gather everything on one side, so we get a quadratic equal to 0. We'll factor that, it's very easy to factor, and we get two roots, 1 and 6. Now normally with algebra, you'd think okay, we must be done, we found the value of X. 但是利用自由基方程，我们必须小心。我们是否知道这两个根源都在工作？
Maybe they both do, or maybe one of them is an extraneous root. So we have to check our answers. We have to check each answer we found to make sure they worked, because right, now just looking at them 1 and 6, we don't know. Are those both true roots, are they both extraneous roots? Do they work in the original equation?
The only way we find is by plugging them in. So, here's the original equation. Here are the roots that we found from the algebra. So first of all we're going to check the first one, x equals 1. 将其插入左侧，我们得到1加3平方根4的平方根，它为2.将其插入右侧，我们得到1减3，为负2。
So the two sides of the equation are not equal. One side equals 2, one side equals negative 2. 所以这个根不起作用。现在，我们将检查另一个。 Plug it in to the left side, we get square root of 6 plus 3, of course that's 9. Square root of 9 is 3.
On the other side we get, 6 minus 3, which is also 3. The two sides work, so that one does legitimately work. And it does solved the problem. So, this equation has one solution that works, x equals 6. 这是唯一有效的解决方案。x等于无关根，因为即使 我们跟着代数正确,即使the algebra gave us that root, that root does not actually work in the original equation.
We need to square both sides to undo the radical, but this very act can produce extraneous roots. If we get a quadratic after squaring, which is common on the test, the algebra will lead to two roots. 有时两个根都是工作。有时一个根作用，一个是无关紧要的。
Sometimes both are extraneous, and the equation has no solution. So, here's a practice problem. Pause the video, and then we'll talk about this. 好的。所以，在这里，我们在双方都有激进的。
Radical equals radical. So of course, we're just gonna square both sides. 我们得到2x减去2等于x减去4.嗯，非常简单的方程来解决。 And we get x equals negative 2. All right, very good.
但now, what happens here if we plug this back into the original equation? When we plug this in, this results in the square root of a negative on both sides. 因此，我们得到了负6的平方根，负6的平方根是实数系之外的东西，它不会在数字线上的任何地方。 所以我们不能用它来做数学。这只是为了我们的目的，这只是一个错误和 该等式没有解决方案。
Finally, keep in mind that we should square both sides only when the radical is by itself on one side of the equation. If the radical appears on, with other terms on one side, we will have to isolate the radical on one side, before it would make sense to square both sides. So, here's a practice problem video, and then we'll talk about this.
Okay, so we do not have the radical by itself. So the very first thing we have to do, is subtract that 2 from both sides. So we get the radical, 4 minus 3x equals x minus 2. 现在我们可以平方两侧。当然，我们得到了差异的平方。
The square of that binomial. And that expands out to x squared minus 4x plus 4. 现在,我们要两边同时减去4,dd 3X to both sides and this will lead us to x squared minus x. 非常容易因素，对X次x减去1.而代数导致我们的解决方案x等于0和x等于1。
现在我们需要检查这些答案。 好的。So those are the roots that the algebra found for us. First of all, check x equals 0. Plug this into the left side, and 我们得到的是2加根4，这是2加2，即4。
将其插入右侧，它是0.当然，4不等于0。 So this one does not work. So this would be an extraneous root. 现在检查x等于1.将其插入左侧，我们得到2加4.3倍1，所以4减3。
And of course, that would be 1. And so, that's gonna be 2 plus 1 which is 3. 当然，这不等于1.不等于x，在等式的另一侧是1。 So this one doesn't work either. And so, neither one of the roots that the algebra gave us works.
So this equation simply has no solution. Both the solutions that the algebra gave, were extraneous roots. In summary, to undo a radical equation, we need to square both sides. We have to mo, move something else to the other side sometimes, 在平方之前隔离激进术。换句话说，我们本身需要激进的。
So there are other terms on that side with the radical. We need to get rid of them, move them to the other side before we can square. And the very act of squaring produces extraneous roots, therefore, we must check each answer the algebra gives us back in the original equation.
