Quadratic Equations — everyone’s best friend! Now we’re learning about some deeper concepts of algebra. So let’s go over the basics.
Transcript: Quadratic Equations
Quadratic equations. You may remember quadratics from our discussion of expressions and factoring. A quadratic equation is one of the form, ax squared + bx + c = 0. In other words, a quadratic expression set equal to 0.
Most often these equations have two different solutions. So before we even begin to talk about how to solve this, let me say, that the strategy that we learned from linear equations trying 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 to 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.
Majority of Quadratic Equations
There’s a completely different procedure of equation solving that we follow, and that’s what we’re gonna talk about here. The vast majority of quadratic equations you will see on the test will have two 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’ll see on the test can be solved by the factoring methods we’ve 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’re going to employ all those factoring strategies here.
We will factor a quadratic to a product of linear binomial. And notice we’ll be doing this so that this product equals 0. So very important to get things equals to 0 first, then to 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 zero, if a product of two things equal zero, then either one is zero or the other is zero.
Before we go on let’s think about this statement for a minute. Keep in mind, that the “or” that appears in that statement is the mathematical or, not the or of colloquial language. What’s the distinction of drawing here? The mathematical OR is an inclusive or. That is it includes the and case. Thus, A = 0 or B = 0 includes three cases.
It could be that A = 0 and B doesn’t, it could be that B = 0 and A doesn’t, or could be that they both equal 0 at the same time. You see sometimes in colloquial language people use the word or to mean the exclusive or. 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, and this is important here and it will become important again in the probability lessons.
Okay, now we are ready to solve. So the first thing we’re gonna do is factor, and we can do that from our factoring lessons.
Now, we’re going to apply this zero product property. So if we’re multiplying two things to get zero, it means either one equal zero or the other equal zero. Now, we have too many equations. We’ve solve them in parallel, and we get that x = -12, or x = 2. And those are the two solutions to that original equation.
Here is another example, now before we can apply the 0 product property, we have to have something that equals zero. Our first step is gonna be to get the quadratic equal to 0, we are going to do this by subtracting eleven from both sides. So we get x squared- 3x- 54 = 0. So 54, and that’s a tricky number, but notice 6 x 9 = 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 0, 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. This looks a little bit trickier, but we’ll start by subtracting 13 from both sides. And now notice we have a greatest common factor of 5. So, we could factor that out and in fact what we can just do is divide both sides by 5, because the gross 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. 0 product property and solve for the individual roots, and those are the answers. Suppose we have this. Now, we have a quadratic equaling a quadratic.
Looks very tricky, and of course the test loves to throw stuff like this at you. But all you have to do is just get everything on one side. So, we are gonna subtract the x squared, subtract the the 10x, and subtract the 16. And this will get us down to x squared- 6x + 9 = 0, and notice that’s a special pattern. That is the square of a difference.
So, we can write that immediately as (x-3) (x-3). It means that our only possibility forgetting 0 of x- 3 = 0 or another word of x = 3. So, this is an example of a quadratic that has one solution instead of two.
Here’s a Tricky One
Now this one, 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, if we square positive we get a positive number, we square zero we get zero, we square negative we get a positive, but there is nothing we can square to get a negative, this is impossible.
This is a simple example of a quadratic that has no solution. So as I said at the beginning, most quadratic have two solutions–but there are some that have one, and then they’re some that have none.
And this is a simple example of a quadratic with no solution.
More Typical — Follow the Procedure
For the vast majority of quadratic on the test, you will follow this procedure. First of all, you’ll get everything on one side of the equation set equal to zero. Then you will divide off any greatest common factor. You will factor it into a product of linear binomials. And then use the Zero Product Property to create two linear equations, and solve each one of them separately.
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 Quantitative section, you may need to know and use the Quadratic Formula.
We will cover that in a lesson in the Powers and Roots module. So, don’t worry about that now, we’ll get to that later. To solve most quadratic equations on the test. First of all you need to gather all terms to one side set equal to zero. Then divide by any numerical greatest common factor. Then factor into a product of two linear binomials and use the the Zero Product Property to separate into two linear equations, and solve.