Now we'll talk about the quadratic formula. Now of course this is out of place in some sense, because mostly when we're talking about quadratics, that was back in the algebra section. We were talking about factoring. Usually if you have to solve a quadratic, the best method will be factoring it. But sometimes quadratic equations cannot be factored and at least one option for these is to use the Quadratic Formula.

There's another method called completing the square which I've demonstrated in a few lessons. 如果二次异常类似于二项式的平方，并且您应该知道这些模式。 总和的平方，差异的平方。如果它接近其中一个模式，那么解决它通常很容易 without the quadratic formula.

However, there are some quadratics that are not factorable, and you can't really fit it very easily into one of those neat algebraic formula, and 然后二次配方通常是你最好的选择。因此，这是二次公式。 First of all, keep in mind it is an if then statement. If ax squared plus bx plus c equals zero.

所以请注意，二次方程式设置等于零，所以这是标准形式。如果我们将二次划分为标准形式并读取系数， then the solution for x will follow that familiar pattern, that familiar formula. Notice also there's a plus-minus sign in that formula. 通常，我们会得到两个根。请记住，二次是抛物线的图表， many parabolas intersect the x axis twice, so that's why you'd get two roots.

你没有，不需要记住这个公式。如果您需要，测试将始终提供此公式。 And once again, most of the time, with a quadratic your best bet is to factor it, you don't need this formula. 如果某些事情完全不可搞令，您唯一的时间，此测试将为您提供此公式。

So, just keep that in mind. In particular, notice the expression under the radical in the Quadratic Formula, B平方减去4AC。这有时称之为判别。 这是一个你不需要知道这一行为的术语。但这种表达，B平方英尺4AC，非常重要。

The reason is, if it's positive, then we get two real square roots, and then we're going to wind up with two different values for x. It's going to be negative b plus something, and negative b minus something over 2A. 我们将获得两个根。在极少数情况下，当B平方减去4AC等于0， then the quadratic has one real root, and you'll notice for any perfect square, if you go back and look at the square of a sum or square of a difference from the algebra section, all of these obey this condition.

That b squared minus 4ac equals zero. So this would be a parabola that is simply tangent to the X axis at its vertex, 所以它只有一个解决方案。当然，这种表达也可以是负面的。 Well think about that. If it's negative, this is something under the square root, so we have square root of a negative.

That would be an imaginary number. We'd get two imaginary solutions. And of course as always, what you'd get is some real number plus or minus some imaginary number. The two roots would be two complex conjugates. It is very unlikely that the ACT is going to give you a Quadratic Formula 这将结束有一个虚构的根，但可能会发生。

所以它只是要记住的事情。这是一个非常简单的例子。 So there's a Quadratic Formula. It is quadratic equation. 它已经是标准形式。它已经设置为等于零。

我们要解出x。它不是立即obvious how we would factor that or use completing the square, so the quadratic formula is actually not a bad choice with this equation. So we can see that a equals one, b and c equal negative one, 1. 当您读取二次配方的B和C时，非常重要。

So the quadratic formula which the test would give us. We'd plug in these values under the radical we get a root five. And so we have two roots here. The two roots, one of these is one plus root five over two. 另一个是一个减去两个超过两个。顺便提及，第一个根是黄金比，第二根源是 the reciprocal of the golden ratio, but you do not need to know that for the ACT.

Here's another example. We're going to solve this, not notice this one is not in standard form. So step one is always put things in standard form. So we have to put it in standard form, we subtract 12 from both sides, 我们减去了四个。事实证明，如果我要解决这个问题，我就会 说这是非常，非常接近完成方形问题，我会以这种方式解决，但让我们用二次公式来解决它。

所以我们得到一个等于的，B等于负数12，C等于31.插入所有这些数字，四次31， well, four times 30 is 120, so four times 30, rounded to 124. We subtract that, we get 12 plus root 20 over two. 现在，记住我们对激进的课程，我们可以简化20，因为20是四次五次。

And four is a perfect square, so we can separate that into square root of four times square root of five, and the square root of four is two. Well now everything in the numerator is divisible by tow, so we can cancel the two, and we get six plus or minus root five. And that is the solution to this equation, six plus root five and six minus root five, those are the two roots.

这是一个练习问题。暂停视频，然后我们会谈谈这个。 Okay, so this is in the form that the ACT would state it. They give us the quadratic formula, they state all that very clearly, and then they ask us to solve the problem. Now, of course, they did slip us the trick here.

They gave us an equation that's not in standard form, it's not equal to zero, so we just have to subtract three from both sides and get it equal to zero. 好的。所以现在我们有标准形式的东西。顺便提一下，如果您想解决这个问题，如果您想要完成广场 to add whatever you needed to add to get the perfect, the perfect square, the square of a difference.

这也是一个完全有效的方法来解决它。我刚刚指出，不要觉得被迫使用二次公式， 如果其他解决方案的方法更容易。但我将展示二次公式。 我们插上一切。当然，四次七个是28,36.33.33.

八个平方根可以被简化，因为八个是八次的四倍，八个是四次平方根的平方根 换句话说，两个根两根。然后我们可以将所有东西分为两个，我们得到三加或减去根。 So that's the solution to this particular equation. We go back to the problem and we select answer choice B.

In summary, we can find the solution of an unfactorable quadratic using the quadratic formula. Now I wanna emphasize once again, it's not your only option, and in fact completing the square is often a much quicker, more efficient option. 但你当然可以使用二次公式。当您需要使用二次公式时，该动作将始终提供。

你不需要记住它。您必须确保您正在解决的二次方程是标准 在读取A，B和C之前的表格插入二次公式。