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Quadratic Formula
Transcript
现在我们将讨论二次公式。现在，当然，这在某种意义上是不合适的，因为，主要是在我们的时候
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
至少一个选项是使用二次公式。
There's another method called completing the square, which I have demonstrated in a few lessons. If the quadratic resembles the square of a binomial, and you should know those patterns, the square of the sum, the square of a difference, if it's close to one of those patterns, it's often pretty easy to solve it 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 patterns, so then the Quadratic Formula is often your best bet. So this is the Quadratic Formula. First of all, keep in mind, it is an If, Then statement. If ax squared + bx + c = 0, so notice that is a quadratic equation set equal to 0, so that's in standard form.
If we put that quadratic into standard form and read off the coefficients, then the solution for x will follow that familiar pattern, that familiar formula. 请注意，通常有一个+/-登录的公式，通常会得到两个根。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. You do not, do not need to memorize this formula.
如果您需要，测试将始终提供此公式，并且再次，大多数时间，具有二次，您最好的选择是要考虑它。 You don't need this formula. The only time the test will give you this formula if something is utterly 难以忘记，所以请记住这一点。特别是， 注意在二次公式中的激进下的表达，B平方-4Ac。
这有时被称为判别。这是一个你不需要知道ATC的术语，但是 that little expression, b squared- 4ac, that's very important. The reason is, if it's positive, then we get two real square roots, and then we're gonna wind up with two different values for x. It's gonna be negative b plus something and negative b minus something over 2a.
我们会得到两个根源。在极少数情况下，当B平方 - 4AC = 0时，二次有一个真正的根。 如果您返回并查看与代数部分的差异的总和或平方的正方形，那么您将注意到，请注意， all of these obey this condition that b squared- 4ac = 0. So this would be a parabola that is simply tangent to the x-axis at its vertex, so it has only one solution, and, of course, this expression can also be negative.
Well, think about that. If it's negative, this is something under the square root, so 我们有一个消极的平方根。这将是一个虚构的数字。 我们得到了两个虚构的解决方案，当然，一如既往地，你得到的是一些实数加或减去一些虚构的数字。
这两个根部是两个复杂的缀合物。这行神不太可能会给你一个二次 公式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。 Here's a very simple example. So there's a quadratic equation.
It's already in standard form. It's already set equal to zero, and we're gonna solve for x. 所以它没有立即显而易见的是，我们如何考虑到或使用正方形，所以 二次配方实际上不是这种等式的不良选择。所以我们可以看到a = 1，b和c = -1。
Very important to remember those negative signs when you're reading off a, b and c for the quadratic formula. 因此，测试将为我们提供的二次公式，我们会插入这些值。在激进的下，我们得到一个根5，所以我们这里有两个根。 两个根，其中一个+根5超过2.另一个是1-根5超过2。
Incidentally, that first root is the golden ratio, and the second root is the reciprocal of the golden ratio, but you do not need to know that for the ACT. Here's another example. We're gonna solve this. 现在，请注意这一个不是标准形式，所以第一步是用标准形式把东西放在标准形式上，所以我们必须用标准形式。
We subtract 12 from both sides, we subtract 4. 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 it solve it that way, but let's solve it with the quadratic formula. So, we get a = 1, b = -12, c = 31. Plug in all those numbers, and 4 times 31, well, 4 times 30 is 120, so 4 times 31 is 124.
我们减去了这一点。我们得到12 +根20超过2。 Now remember the lessons that we had on radicals. We can simplify 20 because 20 is 4 times 5, and 4 is a perfect square, 因此，我们可以将其分成4倍平方根5的平方根，平方根4是2。
Well, now everything in the numerator's divisible by 2, so we can cancel the 2, and we get 6 plus or minus root 5, and that is the solution to this equation. 6 + root 5 and 6 + root 5, those are the two roots. Here's a practice problem, pause the video, and then we'll talk about this. 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 0, so we just have to subtract 3 from both sides and get it equal to 0. So now we have something in standard form. Incidentally, if you wanted to solve this with completing the square, if you wanted to add whatever you needed to add to get the perfect square, 差异的平方。
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 if another method of solution is easier for you, but here I'll demonstrate the quadratic formula. We plug everything in. Of course, 4(7) is 28, 36- 28 is 8.
The square root of 8 can be simplified because 8 is 4 x 2, so the square root of 8 is the square root of 4 times square root of 2, or in other words 2 root 2. 然后我们可以用2划分一切，我们得到3 +/- root 2.这就是这种特定方程式的解决方案。 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.
现在我想再次强调，这不是你唯一的选择。事实上，完成广场通常更快， more efficient option, but you certainly can use the quadratic formula. When you need to use the quadratic formula, the ACT will always supply it. 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 form before you read a, b,and c to plug into the quadratic formula.
Read full transcript
There's another method called completing the square, which I have demonstrated in a few lessons. If the quadratic resembles the square of a binomial, and you should know those patterns, the square of the sum, the square of a difference, if it's close to one of those patterns, it's often pretty easy to solve it 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 patterns, so then the Quadratic Formula is often your best bet. So this is the Quadratic Formula. First of all, keep in mind, it is an If, Then statement. If ax squared + bx + c = 0, so notice that is a quadratic equation set equal to 0, so that's in standard form.
If we put that quadratic into standard form and read off the coefficients, then the solution for x will follow that familiar pattern, that familiar formula. 请注意，通常有一个+/-登录的公式，通常会得到两个根。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. You do not, do not need to memorize this formula.
如果您需要，测试将始终提供此公式，并且再次，大多数时间，具有二次，您最好的选择是要考虑它。 You don't need this formula. The only time the test will give you this formula if something is utterly 难以忘记，所以请记住这一点。特别是， 注意在二次公式中的激进下的表达，B平方-4Ac。
这有时被称为判别。这是一个你不需要知道ATC的术语，但是 that little expression, b squared- 4ac, that's very important. The reason is, if it's positive, then we get two real square roots, and then we're gonna wind up with two different values for x. It's gonna be negative b plus something and negative b minus something over 2a.
我们会得到两个根源。在极少数情况下，当B平方 - 4AC = 0时，二次有一个真正的根。 如果您返回并查看与代数部分的差异的总和或平方的正方形，那么您将注意到，请注意， all of these obey this condition that b squared- 4ac = 0. So this would be a parabola that is simply tangent to the x-axis at its vertex, so it has only one solution, and, of course, this expression can also be negative.
Well, think about that. If it's negative, this is something under the square root, so 我们有一个消极的平方根。这将是一个虚构的数字。 我们得到了两个虚构的解决方案，当然，一如既往地，你得到的是一些实数加或减去一些虚构的数字。
这两个根部是两个复杂的缀合物。这行神不太可能会给你一个二次 公式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。 Here's a very simple example. So there's a quadratic equation.
It's already in standard form. It's already set equal to zero, and we're gonna solve for x. 所以它没有立即显而易见的是，我们如何考虑到或使用正方形，所以 二次配方实际上不是这种等式的不良选择。所以我们可以看到a = 1，b和c = -1。
Very important to remember those negative signs when you're reading off a, b and c for the quadratic formula. 因此，测试将为我们提供的二次公式，我们会插入这些值。在激进的下，我们得到一个根5，所以我们这里有两个根。 两个根，其中一个+根5超过2.另一个是1-根5超过2。
Incidentally, that first root is the golden ratio, and the second root is the reciprocal of the golden ratio, but you do not need to know that for the ACT. Here's another example. We're gonna solve this. 现在，请注意这一个不是标准形式，所以第一步是用标准形式把东西放在标准形式上，所以我们必须用标准形式。
We subtract 12 from both sides, we subtract 4. 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 it solve it that way, but let's solve it with the quadratic formula. So, we get a = 1, b = -12, c = 31. Plug in all those numbers, and 4 times 31, well, 4 times 30 is 120, so 4 times 31 is 124.
我们减去了这一点。我们得到12 +根20超过2。 Now remember the lessons that we had on radicals. We can simplify 20 because 20 is 4 times 5, and 4 is a perfect square, 因此，我们可以将其分成4倍平方根5的平方根，平方根4是2。
Well, now everything in the numerator's divisible by 2, so we can cancel the 2, and we get 6 plus or minus root 5, and that is the solution to this equation. 6 + root 5 and 6 + root 5, those are the two roots. Here's a practice problem, pause the video, and then we'll talk about this. 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 0, so we just have to subtract 3 from both sides and get it equal to 0. So now we have something in standard form. Incidentally, if you wanted to solve this with completing the square, if you wanted to add whatever you needed to add to get the perfect square, 差异的平方。
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 if another method of solution is easier for you, but here I'll demonstrate the quadratic formula. We plug everything in. Of course, 4(7) is 28, 36- 28 is 8.
The square root of 8 can be simplified because 8 is 4 x 2, so the square root of 8 is the square root of 4 times square root of 2, or in other words 2 root 2. 然后我们可以用2划分一切，我们得到3 +/- root 2.这就是这种特定方程式的解决方案。 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.
现在我想再次强调，这不是你唯一的选择。事实上，完成广场通常更快， more efficient option, but you certainly can use the quadratic formula. When you need to use the quadratic formula, the ACT will always supply it. 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 form before you read a, b,and c to plug into the quadratic formula.
