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二次公式
成绩单
现在我们来讨论二次公式。当然，这在某种意义上是不合适的，因为在我们谈话的时候
关于二次函数，那是在代数部分。我们说的是保理。
通常，如果你要解一个二次方程，最好的方法就是分解它。但有时二次方程不能被分解，而且至少有一个选择
这是使用二次公式。
还有另一种方法叫做完成正方形，我在几节课中演示过。 If the quadratic resembles the square of a binomial, and you should know those patterns. The square of the sum, the square of the difference. If it's close to one of those patterns it's often pretty easy to solve it 没有二次公式。
然而，有些二次函数是不可分解的，你不能很容易地把它拟合成一个简洁的代数公式，而且 then the quadratic formula is often your best bet. So, this is the Quadratic Formula. 首先，请记住这是一个if-then语句。如果ax的平方加上bx加上c等于零。
So notice that is a quadratic equation set equal to zero, so that's in standard form. If we put that quadratic into standard form and read off the coefficients, 那么x的解将遵循熟悉的模式，熟悉的公式。注意，公式中还有一个正负号。 Typically, we'll get two roots. Remember that the graph of a quadratic is a parabola, and 许多抛物线与x轴相交两次，所以得到两个根。
You do not, do not need to memorize this formula. The test will always provide this formula if you need it. 再说一次，大多数时候，对于二次型，你最好的办法就是考虑它，你不需要这个公式。 The only time that you will, this test will give you this formula if something is utterly unfactorable.
所以，记住这一点。特别注意，在二次公式中，根下面的表达式， b squared minus 4ac. This is sometimes call the discriminant. That's a term that you do not need to know for the ACT. But that expression, b squared minus 4 ac, is very important.
原因是，如果它是正的，那么我们得到两个真正的平方根，然后我们得到两个不同的x值。 它将是负b加上某个值，负b减去2A以上的某个值。 We're going to get two roots. In rare cases when b squared minus 4ac equals 0, 然后二次方有一个实根，你会注意到任何一个完美的平方，如果你回头看一个和或的平方 代数部分的差平方，所有这些都服从这个条件。
b的平方减去4ac等于零。这是一条抛物线，它的顶点与X轴相切， so it has only one solution. And of course this expression can also be negative. 好好想想。如果是负数，这是平方根以下的值，所以 我们有负数的平方根。
那将是一个虚数。我们会得到两个假想的解决方案。 当然，像往常一样，你会得到一些实数加上或减去一些虚数。 两个根是两个复杂的共轭体。法案不太可能给出一个二次公式 that's going to wind up having an imaginary root, but it could happen.
So it's just something to keep in mind. Here's a very simple example. 所以有一个二次公式。它是二次方程。 It's already in standard form. It's already set equal to zero.
我们要解x，现在还不清楚我们该如何考虑这个因素，或者用平方来完成， 所以二次公式实际上是一个不错的选择。所以我们可以看到a等于1，b和c等于负1，1。 Very important to remember those negative signs when you're reading off a, b and c of the quadratic formula.
所以这个测试会给我们一个二次公式。我们把这些值代入根5下。 所以我们这里有两个根。两个根，其中一个是1加上根5加2。 The other is one minus root five over two. Incidentally, that first root is the golden ratio, and the second root is 黄金比率的倒数，但是你不需要知道。
这是另一个例子。我们要解决这个问题，不要注意这个不是标准形式。 所以第一步总是把东西放在标准的形式。所以我们必须用标准形式，两边减去12， we subtract four. As it turns out, if I were gonna solve this, I would actually say that this is very, very close to a completing the square problem, and I would solve it that way, but let's solve it with the quadratic formula.
So we get a equals one, b equals negative 12, c equals 31. Plug in all those numbers, and four times 31, 四乘以30等于120，四乘以30，四舍五入为124。我们减去它，得到12加上根20除以2。 Now, remember the lessons that we had on radicals, we can simplify 20 because 20 is four times five.
四是一个完美的平方，所以我们可以把它分成四的平方根乘以五的平方根，四的平方根是二。 现在分子中的所有东西都可以被2整除，所以我们可以取消2，得到6加或减根5。 这就是这个方程的解，六加五和六减五，这是两个根。
Here's a practice problem. Pause the video and then we'll talk about this. 好吧，这就是法案所说的形式。他们给了我们一个二次公式，他们非常清楚地说明了这一切 然后他们要求我们解决问题。现在，当然了，他们确实给了我们这个把戏。
他们给了我们一个不是标准形式的方程，它不等于零，所以我们只需要从两边减去3，得到它等于零。 好吧。现在我们有一些标准form. Incidentally, if you wanted to solve this with completing the square if you wanted 为了得到完美的，完美的平方，差的平方。
That's also a perfectly valid way to solve it. I'll just point out, don't feel compelled to use the quadratic formula, if an other method of solution is easier for you. But here I'll demonstrate the quadratic formula. We plug everything in. Of course, four times seven is 28, 36 minus 28 is eight.
Square root of eight can be simplified because eight is four times two so square root of eight is square root of four times square root of two or in other words two root two. Then we can divide everything by two and we get three plus or minus root two. 这就是这个方程的解。我们回到问题，选择答案B。
总之，我们可以用二次公式求出不可乘二次方程的解。 现在我想再次强调，这不是你唯一的选择，事实上，完成广场往往是一个更快，更有效的选择。 But you certainly can use the quadratic formula. When you need to use the quadratic formula the ACT will always supply it.
You do not need to memorize it. You must make sure that the quadratic equation you are solving is in standard form before you read a, b, and c to plug in the quadratic formula.
阅读完整的成绩单
还有另一种方法叫做完成正方形，我在几节课中演示过。 If the quadratic resembles the square of a binomial, and you should know those patterns. The square of the sum, the square of the difference. If it's close to one of those patterns it's often pretty easy to solve it 没有二次公式。
然而，有些二次函数是不可分解的，你不能很容易地把它拟合成一个简洁的代数公式，而且 then the quadratic formula is often your best bet. So, this is the Quadratic Formula. 首先，请记住这是一个if-then语句。如果ax的平方加上bx加上c等于零。
So notice that is a quadratic equation set equal to zero, so that's in standard form. If we put that quadratic into standard form and read off the coefficients, 那么x的解将遵循熟悉的模式，熟悉的公式。注意，公式中还有一个正负号。 Typically, we'll get two roots. Remember that the graph of a quadratic is a parabola, and 许多抛物线与x轴相交两次，所以得到两个根。
You do not, do not need to memorize this formula. The test will always provide this formula if you need it. 再说一次，大多数时候，对于二次型，你最好的办法就是考虑它，你不需要这个公式。 The only time that you will, this test will give you this formula if something is utterly unfactorable.
所以，记住这一点。特别注意，在二次公式中，根下面的表达式， b squared minus 4ac. This is sometimes call the discriminant. That's a term that you do not need to know for the ACT. But that expression, b squared minus 4 ac, is very important.
原因是，如果它是正的，那么我们得到两个真正的平方根，然后我们得到两个不同的x值。 它将是负b加上某个值，负b减去2A以上的某个值。 We're going to get two roots. In rare cases when b squared minus 4ac equals 0, 然后二次方有一个实根，你会注意到任何一个完美的平方，如果你回头看一个和或的平方 代数部分的差平方，所有这些都服从这个条件。
b的平方减去4ac等于零。这是一条抛物线，它的顶点与X轴相切， so it has only one solution. And of course this expression can also be negative. 好好想想。如果是负数，这是平方根以下的值，所以 我们有负数的平方根。
那将是一个虚数。我们会得到两个假想的解决方案。 当然，像往常一样，你会得到一些实数加上或减去一些虚数。 两个根是两个复杂的共轭体。法案不太可能给出一个二次公式 that's going to wind up having an imaginary root, but it could happen.
So it's just something to keep in mind. Here's a very simple example. 所以有一个二次公式。它是二次方程。 It's already in standard form. It's already set equal to zero.
我们要解x，现在还不清楚我们该如何考虑这个因素，或者用平方来完成， 所以二次公式实际上是一个不错的选择。所以我们可以看到a等于1，b和c等于负1，1。 Very important to remember those negative signs when you're reading off a, b and c of the quadratic formula.
所以这个测试会给我们一个二次公式。我们把这些值代入根5下。 所以我们这里有两个根。两个根，其中一个是1加上根5加2。 The other is one minus root five over two. Incidentally, that first root is the golden ratio, and the second root is 黄金比率的倒数，但是你不需要知道。
这是另一个例子。我们要解决这个问题，不要注意这个不是标准形式。 所以第一步总是把东西放在标准的形式。所以我们必须用标准形式，两边减去12， we subtract four. As it turns out, if I were gonna solve this, I would actually say that this is very, very close to a completing the square problem, and I would solve it that way, but let's solve it with the quadratic formula.
So we get a equals one, b equals negative 12, c equals 31. Plug in all those numbers, and four times 31, 四乘以30等于120，四乘以30，四舍五入为124。我们减去它，得到12加上根20除以2。 Now, remember the lessons that we had on radicals, we can simplify 20 because 20 is four times five.
四是一个完美的平方，所以我们可以把它分成四的平方根乘以五的平方根，四的平方根是二。 现在分子中的所有东西都可以被2整除，所以我们可以取消2，得到6加或减根5。 这就是这个方程的解，六加五和六减五，这是两个根。
Here's a practice problem. Pause the video and then we'll talk about this. 好吧，这就是法案所说的形式。他们给了我们一个二次公式，他们非常清楚地说明了这一切 然后他们要求我们解决问题。现在，当然了，他们确实给了我们这个把戏。
他们给了我们一个不是标准形式的方程，它不等于零，所以我们只需要从两边减去3，得到它等于零。 好吧。现在我们有一些标准form. Incidentally, if you wanted to solve this with completing the square if you wanted 为了得到完美的，完美的平方，差的平方。
That's also a perfectly valid way to solve it. I'll just point out, don't feel compelled to use the quadratic formula, if an other method of solution is easier for you. But here I'll demonstrate the quadratic formula. We plug everything in. Of course, four times seven is 28, 36 minus 28 is eight.
Square root of eight can be simplified because eight is four times two so square root of eight is square root of four times square root of two or in other words two root two. Then we can divide everything by two and we get three plus or minus root two. 这就是这个方程的解。我们回到问题，选择答案B。
总之，我们可以用二次公式求出不可乘二次方程的解。 现在我想再次强调，这不是你唯一的选择，事实上，完成广场往往是一个更快，更有效的选择。 But you certainly can use the quadratic formula. When you need to use the quadratic formula the ACT will always supply it.
You do not need to memorize it. You must make sure that the quadratic equation you are solving is in standard form before you read a, b, and c to plug in the quadratic formula.
