二次方程
Transcript
二次方程。你可能还记得我们讨论表达式和因式分解时的二次函数。
二次方程的形式是A的平方加上bx加上c等于零。换句话说，二次表达式等于零。
Most often these equations have two different solutions. So before we even begin to talk about how to solve these, let me say,
that the strategy that we learn for linear equations, try to get all the x's on one side of the equation, all the constants on the other side of the equation.
That is exactly the strategy you wanna follow for linear equations, but following that for quadratics is an unmitigated disaster. So do not follow the same strategy. When we're dealing with quadratics is a completely different procedure of 我们接下来要讨论的方程求解。你将在测试中看到的绝大多数二次方程都有两个 solutions and the quadratic coefficient, the coefficient of x squared will equal 1 unless a numerical greatest common factor can be factored out from all three terms.
The vast majority of quadratic equations you will see on the test can be solved by the factoring methods we discussed in the factoring lessons. So if you haven't seen those lessons, it will be very helpful to watch those before watching this video. Because we are gonna employ, all those factoring strategies here. We will factor a quadratic to a product of linear binomials.
注意，我们这样做，这个积等于0。所以先把东西设为0是很重要的。 Then do this factoring. Because once we have a product that equals 0, we can apply this mathematical law, the Zero Product Property. The Zero Product Property says, if A times B equals 0, 如果两件事的乘积等于0。
那么，一个是0，另一个是0。在我们继续之前，让我们先考虑一下这句话。 Keep in mind that the OR that appears in that statement is the mathematical OR. Not the or of colloquial language. 好吧，我在这里画的区别是什么？数学OR是一个包含的OR。
也就是说，它包括和案例。因此，A等于0或B等于0，包括三种情况。 可能是A等于0，B不等于0，也可能是B等于0，A不等于0。 Or it could be that they both equal 0 at the same time. You see sometimes in colloquial language, people use the word or 指排他性的或。
They mean you could do this or that and the implication is that you can't do both. That's the exclusive or. That is not how the word or is used in mathematics. The word used in mathematics is the inclusive or which includes the and case. 这在这里很重要，在概率课上也会变得很重要。
现在，我们可以解决了。所以我们要做的第一件事就是。 And we can do that from our factoring lessons. Now we're going to apply the zero product property. 所以如果我们把两个东西相乘得到0，这意味着要么1等于0，要么另一个等于0。
Now we have too many equations, we solve them in parallel. And we get that x equals negative 12 or x equals 2, and those are the two solutions to that original equation. Here's another example. 现在我们可以应用零产品属性之前,we have to have something that equals zero.
So, our first step is gonna be to get the quadratic equal to zero, and we're gonna do this by subtracting 11 from both sides. 所以，我们得到x的平方减去3x减去54等于零。54是个棘手的数字，但注意6乘以9等于54， if we have a positive 6, and a negative 9 those will add to negative 3. Those are the numbers we need.
Then we apply the zero product property. Set each one of those equal to zero and solve. Again notice that I'm being very careful to write the word OR at each step. Again the word OR is not decoration. The word OR is a valid piece of mathematical equipment here. Suppose we have to solve this.
Hm. This looks a little bit trickier. Well, we'll start by subtracting 13 from both sides. 现在注意我们有一个最大公因数5。所以，我们可以考虑一下。 In fact, what we could just do is divide both sides by 5. Because, of course, dividing 0 by 5 is just 0.
So we divide by the greatest common factor. Then we get down to something like this, this is something we can factor. Zero product property and solve for the individual roots and those are the answers. 假设我们有这个，现在我们有一个二次等于一个二次的，看起来非常棘手，当然测试喜欢向你扔这样的东西。
But all you have to do is just get everything on one side. So we're gonna subtract the x squared, subtract the 10x, and subtract the 16. And this will get us down to, x squared minus 6x plus 9 equals 0. And notice that's a special pattern. That is a square of a difference. So we can write that immediately as x minus 3 times x minus 3, which means that our only possibility for getting 0 is if x minus 3 equals 0, or in other words, if x equals 3.
So this is an example of a quadratic that has one solution instead of two. Now this one hm,. That's tricky. If we move the 5 to the other side of the equation, we see a problem. There's no way that we can square something and get a negative number. We can square.
如果我们把一个正数平方，我们得到一个正数，我们把0平方，我们得到0，我们把一个负数平方，我们得到一个正数，但是没有什么可以平方得到一个负数。 这是不可能的。这是一个没有解的二次曲线的简单例子。 所以，正如我在一开始说的，大多数二次函数都有两个解，但是有些有一个，有些没有。
这是一个没有解的二次方程的简单例子。 对于测试中的绝大多数二次曲线，您将遵循以下步骤。首先，你会得到方程组一边的所有东西都等于零。 然后你就可以划分出任何最大的公因数。你会把它分解成线性二项式的乘积。
然后利用零积性质建立两个线性方程组，分别求解。 So this is the procedure. Here are some practice problems. Pause the video and solve these on your own and then we'll talk about these. And these are the solutions.
That covers most of what you need to know about quadratics for the test. On very advanced portions of the Quadrat, 在定量部分，你可能需要知道并使用二次公式。我们将在电源和根模块的课程中介绍这一点。 So don't worry about that now. We'll get to that later.
在测试中解决大多数二次方程。首先，你需要把所有的项集中在一边，设为0。 Then divide by any numerical greatest common factor. Then factor into a product of two linear binomials. 并利用零积性质将其分解为两个线性方程组进行求解。
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That is exactly the strategy you wanna follow for linear equations, but following that for quadratics is an unmitigated disaster. So do not follow the same strategy. When we're dealing with quadratics is a completely different procedure of 我们接下来要讨论的方程求解。你将在测试中看到的绝大多数二次方程都有两个 solutions and the quadratic coefficient, the coefficient of x squared will equal 1 unless a numerical greatest common factor can be factored out from all three terms.
The vast majority of quadratic equations you will see on the test can be solved by the factoring methods we discussed in the factoring lessons. So if you haven't seen those lessons, it will be very helpful to watch those before watching this video. Because we are gonna employ, all those factoring strategies here. We will factor a quadratic to a product of linear binomials.
注意，我们这样做，这个积等于0。所以先把东西设为0是很重要的。 Then do this factoring. Because once we have a product that equals 0, we can apply this mathematical law, the Zero Product Property. The Zero Product Property says, if A times B equals 0, 如果两件事的乘积等于0。
那么，一个是0，另一个是0。在我们继续之前，让我们先考虑一下这句话。 Keep in mind that the OR that appears in that statement is the mathematical OR. Not the or of colloquial language. 好吧，我在这里画的区别是什么？数学OR是一个包含的OR。
也就是说，它包括和案例。因此，A等于0或B等于0，包括三种情况。 可能是A等于0，B不等于0，也可能是B等于0，A不等于0。 Or it could be that they both equal 0 at the same time. You see sometimes in colloquial language, people use the word or 指排他性的或。
They mean you could do this or that and the implication is that you can't do both. That's the exclusive or. That is not how the word or is used in mathematics. The word used in mathematics is the inclusive or which includes the and case. 这在这里很重要，在概率课上也会变得很重要。
现在，我们可以解决了。所以我们要做的第一件事就是。 And we can do that from our factoring lessons. Now we're going to apply the zero product property. 所以如果我们把两个东西相乘得到0，这意味着要么1等于0，要么另一个等于0。
Now we have too many equations, we solve them in parallel. And we get that x equals negative 12 or x equals 2, and those are the two solutions to that original equation. Here's another example. 现在我们可以应用零产品属性之前,we have to have something that equals zero.
So, our first step is gonna be to get the quadratic equal to zero, and we're gonna do this by subtracting 11 from both sides. 所以，我们得到x的平方减去3x减去54等于零。54是个棘手的数字，但注意6乘以9等于54， if we have a positive 6, and a negative 9 those will add to negative 3. Those are the numbers we need.
Then we apply the zero product property. Set each one of those equal to zero and solve. Again notice that I'm being very careful to write the word OR at each step. Again the word OR is not decoration. The word OR is a valid piece of mathematical equipment here. Suppose we have to solve this.
Hm. This looks a little bit trickier. Well, we'll start by subtracting 13 from both sides. 现在注意我们有一个最大公因数5。所以，我们可以考虑一下。 In fact, what we could just do is divide both sides by 5. Because, of course, dividing 0 by 5 is just 0.
So we divide by the greatest common factor. Then we get down to something like this, this is something we can factor. Zero product property and solve for the individual roots and those are the answers. 假设我们有这个，现在我们有一个二次等于一个二次的，看起来非常棘手，当然测试喜欢向你扔这样的东西。
But all you have to do is just get everything on one side. So we're gonna subtract the x squared, subtract the 10x, and subtract the 16. And this will get us down to, x squared minus 6x plus 9 equals 0. And notice that's a special pattern. That is a square of a difference. So we can write that immediately as x minus 3 times x minus 3, which means that our only possibility for getting 0 is if x minus 3 equals 0, or in other words, if x equals 3.
So this is an example of a quadratic that has one solution instead of two. Now this one hm,. That's tricky. If we move the 5 to the other side of the equation, we see a problem. There's no way that we can square something and get a negative number. We can square.
如果我们把一个正数平方，我们得到一个正数，我们把0平方，我们得到0，我们把一个负数平方，我们得到一个正数，但是没有什么可以平方得到一个负数。 这是不可能的。这是一个没有解的二次曲线的简单例子。 所以，正如我在一开始说的，大多数二次函数都有两个解，但是有些有一个，有些没有。
这是一个没有解的二次方程的简单例子。 对于测试中的绝大多数二次曲线，您将遵循以下步骤。首先，你会得到方程组一边的所有东西都等于零。 然后你就可以划分出任何最大的公因数。你会把它分解成线性二项式的乘积。
然后利用零积性质建立两个线性方程组，分别求解。 So this is the procedure. Here are some practice problems. Pause the video and solve these on your own and then we'll talk about these. And these are the solutions.
That covers most of what you need to know about quadratics for the test. On very advanced portions of the Quadrat, 在定量部分，你可能需要知道并使用二次公式。我们将在电源和根模块的课程中介绍这一点。 So don't worry about that now. We'll get to that later.
在测试中解决大多数二次方程。首先，你需要把所有的项集中在一边，设为0。 Then divide by any numerical greatest common factor. Then factor into a product of two linear binomials. 并利用零积性质将其分解为两个线性方程组进行求解。
