How to Use the Quadratic Formula
Using the Quadratic Formula
In this video I want to take some time to apply the quadratic formula to a few examples and look at a few special cases as well. Let’s look at the three examples I have up here.
First thing let’s do is let’s write the quadratic formula on the board so we have it for reference. The quadratic formula is x equals negative b, plus or minus the square root of b squared, minus 4ac over 2a.
Where a is the coefficient of x squared, b the coefficient of x, and c is the constant. It’s important that when you’re setting up your quadratic formula that your constant c is on the left side of the equation symbol and you have a zero on the right.
Otherwise, you’re going to get the wrong numbers if you plug in c. We have all these set up in the right form, so let’s go ahead and directly apply the quadratic formula here. For this first one, we have x equals b is negative 4, so negative negative 4 plus or minus the square root, negative 4 squared minus 4, times 1 times 4, over 2 times 1, which is 2.
We can go ahead and simplify now. Negative 4 is 4, plus or minus the square root of 16, minus 4 times 4 is 16, over 2. In this case we have our term, under the, under the radical symbol here. 16 minus 16, this is equal to zero.
The plus or minus doesn’t matter. All that we’re left with is 4 divided by 2 or 2. This equation here only has one solution of x equals 2. Now the second example we have x squared plus 4x minus 12. Our a is 1, our b is 4, and our c is negative 12.
We have x equals negative 4, plus or minus the square root of 4 squared, minus 4 times 1 times 12, divided by 2. Now we can go through and simplify this term and that will give us negative 4 plus or minus the square root of 16, minus 48.
I’m sorry, this is a negative 12 here. 16 plus 48, divided by 2. 16 plus 48 is 64. The square root of 64 is 8. This gives us negative 4 plus or minus 8, divided by 2, or negative 4 plus 8 is 12, or is 4 divided by 2 is 2. If this is a plus our solution is 2, negative 4 minus 8 is negative 12, divided by two is negative 6.
Here we have two solutions, x equals 2 and x equals negative 6. This is the most common solution -or the most common type of solution- you will have with quadratic formulas. Now let’s look at the third example here, x squared plus 4x, plus 12.
Here what we’ll have is x equals negative 4 plus or minus the square root of 4 squared, minus 4 times 1 times 12, divided by 2. Now to simplify this we get negative 4 plus or minus the square root of 16 minus 48, divided by 2.
Now you’ll notice here since the second term here is greater than the first and it’s a minus. What we’re going to end up with here is a negative term underneath the radical. If you write this out you get to this point and you see that this term is greater than that term and there’s a negative sign here, what you can say for certain at that point is that there are no real solutions to this equation.
There will be some complex solutions involving square roots of negative numbers, and so I’ll go ahead and solve it as though we’re taking those into effect, or into account. If you’re looking for just real numbers, if you get to this point and see this sort of thing here where you’re going to wind up with a negative number under the radical you can just stop and say that there are no real solutions.
Let’s proceed to solve it. What we’ll get is 16 minus 48 is negative 32. We’ll have negative 4 plus or minus the square root of negative 32, divided by 2. Now, 32 is 2 raised to the 5th power. We can actually pull out 2 to the 4th or 16, and that gives us negative 4 plus or minus 4 times the square root of negative 2, divided by two, or negative 2 plus or minus 2 times the square root of 2, negative 2.
If we’re looking for complex solutions, this is a solution to this equation. This is just three different examples of applications of the quadratic formula. These are three things you may frequently see. You may see a single real solution, a pair of real solutions, and also the possibility of having no real solutions just complex solutions.