This is another math explainer to help you boost your score in your math class, improve your understanding and hopefully make math a lot less stressful. We're going to talk about how to use the quadratic formula. If you don't know already, quadratic equations are equations that are in the form: ax squared plus bx plus c. When you see that x squared as your highest powered term, that tells you that it's a quadratic or a second degree equation. What those look like when you graph them is they look like parabolas. They look like these u-shaped graphs. What you’re doing when you set this equal to 0 is you're actually looking at where the graph crosses the x-axis.
That's where the y coordinate is equal to 0. Sometimes when you solve these problems, there's a lot of different ways to do it. You can complete the square. You can graph the parabola and see where it crosses the x-axis. You can factor it and set the factors to 0, but sometimes especially with that last example the factoring might be very difficult, or might just not be able to be factored with integers. So this is another possibility that you can use to solve these equations, and it's called the quadratic formula. So you can see here, x equals negative b plus or minus the square root of b squared minus 4ac over by 2a. So what you can see here with these parabolas is that sometimes you're going to get two solutions.
Sometimes you're only going to get one solution where it just reaches the vertex there. Sometimes you're even going to get where it doesn't cross the x-axis at all and there's no solution. Basically what these correspond to is that if this quantity here comes out to zero, you're just going to have negative b over 2a. However, if this quantity comes out to a negative number, you can't take the square root of a negative number without getting imaginary numbers. So you would get no real solutions that are not crossing the x-axis at all through their graph. If it's a positive of this quantity here -- the b squared minus 4ac-- then you would have two solutions. So let us show you what we mean. Let's get into some examples. So this first one here - you can actually factor this one but let's just go ahead and take a look at it. So you can see that a value is one, the b value is one and the c value is negative twelve. So you want to write it in descending order like that so you can identify the a,b and c, and you want to set it equal to zero. If we do our quadratic formula right here, we've got negative b or the opposite of b which is negative one plus or minus the square root of 1 squared minus 4 times 1 times c which is negative 12. You need to make sure that you capture that negative sign - that minus sign is a negative all divided by 2 times 1. So all we have to do now is simplify. So what we have here is negative 1 plus or minus the square root of 1 minus a negative 48 all divided by 2. Now when you subtract a negative, it's like adding a positive. This becomes 49 and the square root of 49 is 7.
Now we have two problems: one where you're adding and one where you're subtracting. So negative one plus seven equals 6 divided by 2 which is 3, and negative one minus seven is negative 8 divided by two which equals negative 4. So all this tells us is that if we were to graph the parabola, it would cross the x-axis over here at 3 and at negative 4. So it looks something like that. Let's look at one more example. Here a value is 6 the b value is negative 9, and the c value is negative 15. So we're going to use the quadratic formula because it looks like this might be a little bit trickier to factor. Of course, here you could factor out a 3 first, but let's just work with it as it is. So we have negative b which is 9 plus or minus the square root of b squared which is 81 minus 4 times 6 times negative 15 4ac all divided by two times six. Now you want to make sure when you see a minus sign you treat that as a negative number. You want to capture that negative sign. If we simplify now, there’s another thing we want to point out. When you have a negative and a minus sign, you've got two negatives. This actually becomes a positive. It's like 81 plus these three integers multiplied together. So this is going to be nine plus or minus the square root of 81 plus 24 times 15 all divided by 12. We’re going to go to the calculator here briefly just to see what these come out to.
It's a little bit of a larger number here. So 24 times 15 is 441, square root of 441 is 21. So we have 9 plus or minus 21 all divided by 12, and you just have to do the two problems separately. 9 plus 21 is 30 over 12. Divide those both by 6 so we get 5 halves. That's one solution, and 9 minus 21 which is negative 12 over 12 which equals negative 1. So these are our two solutions meaning if we put them back into the equation, they'll give us zero. Graphically what it represents is where these points are on the x axis where the graph crosses the x axis.