Hi and welcome. My name is Anthony Varela. And today, we're going to be talking about the quadratic formula. So what is the quadratic formula? When can we use it? Those are the questions we're going to answer today. And then we're going to go through an example or two.
So first, I'd like to talk about how we can solve quadratic equations. And one way is to solve by factoring. So if we have x squared plus 3x plus 2 equals 0, one thing we can do is look for two integers-- we can call them p and q-- such that when we add p and q together, we get this coefficient of the x-term, but when we multiply them together, we get our constant term.
So I can factor this as x plus 2 times x plus 1. We have these two integers, p and q, which is 2 and 1-- added together is this x-term coefficient. But multiplied together, it equals the constant term.
And when we have this written factored, we can set each of our factors equal to 0 and solve for x. So solutions to this quadratic equation is at x equals negative 2 and x equals negative 1.
But what if we have a quadratic equation that's x squared plus 5x plus 1? I cannot identify two integers that multiply together to that constant term, 1, but add together to get 5, the x-term coefficient.
So does this mean then that there are no solutions to this quadratic equation? Well, if we plot it on the graph, we see that there are two x-intercepts. So this equation has two solutions.
So how could I solve this equation? I know I can't do it by factoring. So that's where the quadratic formula comes in handy because we can use this to solve any quadratic equation. So it's particularly useful when an equation cannot be factored or if you just don't know how. You're getting frustrated. You're not quite sure how to factor. You can always use the quadratic formula.
So what is the quadratic formula? Well, I'm going to clear my singing voice because a lot of people love to remember this by singing it to the tune of "Pop Goes the Weasel." So our quadratic formula is (SINGING) x equals negative b plus or minus square root of b squared minus 4ac, all over 2a.
So we see that we have these variables a, b, and c in our quadratic formula. Where do these come from? Well, that's the values of these coefficients in our standard form of a quadratic equation set equal to 0. So a is the coefficient of x squared term. B is the coefficient of the x-term. And c is that constant term.
Now, it's very important that you express this set equal to 0. So if this were a non-zero number, you would have to shift some things around so that it's set equal to 0. That would affect what c is.
So before we get on to using the quadratic formula, I'd like to point out this expression that I've highlighted in red. It's underneath that square root sign in our quadratic formula. This has a special name. We call it the discriminant. And this is special because it helps us determine if a quadratic equation has real solutions.
So because it's underneath a square root sign, it has to be a non-negative in order to evaluate to a real number. So if b squared minus 4ac is greater than or equal to 0, we're going to get real solutions to our quadratic equation, at least one real solution. If b squared minus 4ac is negative or less than 0, we're going to get no real solutions. There's going to be what we call imaginary solutions that we're not going to get into right now.
So remember that b squared minus 4ac is important because if we have a non-negative, we have real solutions. If we do have a negative as our discriminant, we're going to have no real solutions to the equation.
So now let's use the quadratic formula to solve 2x squared plus 4x minus 1 equals 2. Now, this is important. This equation is not set equal to 0. So what I have to do before I grab my values for a, b, and c is subtract 2 from both sides of this equation.
So that changes my c-value here. It's now a negative 3 instead of a negative 1 because I had to subtract 2. So now I see that a equals 2, b equals 4, and c equals negative 3. We're going to plug those values into our quadratic formula.
So here is x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a. So basically, we just need to simplify what looks to be pretty messy, to be very honest. So the easiest thing to do is simplify that denominator. We know that's 2 times 2. So you have a denominator of 4.
Well, our numerator then is negative 4 plus or minus the square root of 4 squared, which is 16. Now we have minus 4 times 2 times negative 3. So 4 times 2 is 8-- times negative 3 is negative 24.
We're subtracting a negative number. So I'm going to write this as addition. So we have 16 plus 24 underneath the square root. So this simplifies to x equals negative 4 plus or minus the square root of 40, all over 4.
So right now, I've confirmed that this equation is going to have at least one real solution because our discriminant is non-negative. But let's simplify that radical. So I can rewrite the square root of 40 as 2 times the square root of 10 because I broke 40 down into 4 times 10, bringing the 4 outside of the radical so that I have 2 because that's the square root of 4. And notice that I can divide everything by 4. That's my-- or my denominator is 4. So I have negative 1 plus or minus the square root of 10 over 2.
So now to find actual x-values, numbers I can make sense out of, I'm going to evaluate negative 1 minus the square root of 10 over 2 and then negative 1 plus the square root of 10 over 2. And rounding to the nearest hundredth, I have x equals 0.58 and x equals negative 2.58. So those are my solutions to my quadratic equation that I wouldn't be able to find by factoring.
We're going to go through one more example of using the quadratic formula. Here we have 3x squared plus 2x plus 1 equals 0. It's set equal to 0. So we know that a equals 3. B is 2. And c is 1. So plugging those into our quadratic formula, we have x equals negative 2 plus or minus the square root of 2 squared minus 4 times 3 times 1, all over 2 times 3.
So we know our denominator is 6. Let's go ahead and simplify our numerator. We have negative 2 plus or minus the square root of 2 squared, which is 4, minus 4 times 3 times 1. So that'd be minus 12.
Well, 4 minus 12 is a negative 8. And we can stop right here. We have a negative number as our discriminant. So we know that this equation has no real solutions.
So let's review our tutorial on the quadratic formula. Well, the quadratic formula can be used to solve any quadratic equation. It's particularly useful when an equation cannot be factored or you don't know how to factor it. And that magic quadratic formula is x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a.
And we talked about this discriminant, this expression b squared minus 4ac. If it's a non-negative number, we have at least one real solution. If it's a negative number, we have no real solutions.
So thanks for watching this tutorial on the quadratic formula-- hope to see you next time.