Hi, this is Anthony Varella, and today, we're going to be solving quadratic equation. So we're going to talk about solutions to quadratic equations. They're also called roots and zeros.
We're going to solve equations written in factored form, and then we're also going to solve equations that have missing terms. That would be coefficients of zero in our general ax squared plus bx plus c. So first, let's talk about what a solution to a quadratic equation is.
Well, they're also called roots or zeros. And we call them zeros, because these are our x values that make y equal zero. So in other words, they're x-intercepts. So here, we have y equals x squared minus x minus six, and we can see that there are two x-intercepts. So this equation happens to have two solutions.
Now depending on where this parabola is on the graph, you might only have one solution. You might not have any instances of x-intercepts, so we would say, that doesn't have any real solutions. There are things called imaginary solutions that we're not going to get into right now.
So when we're solving quadratic equations, we're solving for x when y equals zero. So we'd be solving zero equals x squared minus x minus six. Now looking on the graph, we can quite easily see that our solutions, our roots would be at x equals negative two right here and x equals three right here.
But let's go ahead and solve some quadratic equations algebraically. Our first example, we're going to be solving a quadratic in factored form. Now this is really neat due to the zero factor property. And what this means is that, if a factor equals zero, that means that the entire expression equals zero.
So what we're going to do is we're going to take zero equals x plus two times x minus three, and we're just going to break this up into two equations. We're going to set x plus three equal to zero and x minus three equal to zero. And we'll solve for each factor at equals zero, and this is very straight.
We can just subtract two on both sides to reveal that x equals negative three. And for our other factor, we'll add three to both sides to reveal that x equals three, and those are our two solutions. Let's go over some other examples. This is a quadratic that doesn't have a constant term, so it's c value is zero.
We just have zero equals 3x squared minus 2x. Now this isn't that difficult, because both of our terms on the right side of the equation share a factor of x. So let's factor that out. So if there's no constant term, just factor out x, so we have x times the quantity 3x minus two.
So once again, we can separate this then into two different equations, zero equals x and zero equals 3x minus two. Now one of those is already solved for us. We know that one of our solutions is x equals zero. How about our other solution?
Well, this is just solving a linear equation, right? So we will add two to both sides, so now we have two equals 3x. And then we'll divide both sides by three, so x equals 2/3. That's our other solution here.
Our next example, we're going to be solving a quadratic with no x term. So here, we have zero equals 5x squared minus eight. We have a zero coefficient to our x term, and what we're going to do here is just solve this like a multi-step equation. We're going to apply inverse operations to isolate that variable term, and then we'll apply a square root to solve for x.
So I'm going to add eight to both sides of this equation, so now, I have eight equals 5x squared. To isolate the x squared, I'm going to divide both sides by five. So right now, I have that x squared equals 8/5, which I am just going to say is 1.6. And now, I'm going to apply a square root to both sides.
But when you're applying the square root, remember to include your plus or minus. So x equals positive square root of 1.6, and it also equals negative square root of 1.6. Because a negative times a negative is a positive, so those are our two solutions. We have x equals positive square root of 1.6 and x equals negative square root 1.6.
So let's review solving quadratic equations. We talked about how solutions are also called roots or zeros. These are x values that make y equal zero. So on a graph, those are the x-intercepts. When solving a quadratic in factored form, just solve for each factor equal to zero.
If you have no constant term, that means you can factor out an x and use our same strategy as before. And if you have no x term, you want to isolate the x squared term and apply your square root. But remember to include your plus or minus. So thanks for watching this tutorial on solving quadratic equations. Hope to see you next time.