[MUSIC PLAYING] Let's look at our objectives for today. We'll start by looking at quadratic equations. We'll then talk about the quadratic formula. And finally, we'll do some examples solving quadratic equations with the quadratic formula.
Let's start by looking at quadratic equations. Quadratic equations can be written in the form ax squared plus bx plus c equals 0. Factoring or variable isolation may be used to solve some but not all quadratic equations. However, the quadratic formula can be used to find solutions to all quadratic equations even when factoring or variable isolation is difficult or impossible. Therefore, sometimes, it's necessary to use the quadratic formula to find solutions to a quadratic equation.
Now, let's introduce the quadratic formula. For an equation ax squared plus bx plus c equals 0, the quadratic formula says the solutions for x are equal to negative b plus or minus the square root of b squared minus 4 times a times c all over 2a. The values for a, b, and c in the formula come from the values of a, b, and c in the quadratic equation you want to solve.
Notice that the equation must be set equal to 0 in order to get correct values for a, b, and c. When we use the quadratic formula, we often need to simplify square roots. And this symbol is a plus/minus symbol, which indicates that you must evaluate the expression twice-- once using addition and once using subtraction-- which will lead to two solutions.
Now, let's do an example solving a quadratic equation with a quadratic formula. Suppose Jason is competing in a shot-put event. The height of the shot is modeled by the equation below where x is the time in seconds after the toss. When will the shot hit the ground?
In the equation here, h of x is the height of the shot-- the vertical height-- and x is the time in seconds. We want to find when the shot hit the ground, which means we want to know when the height is 0.
Therefore, the equation we want to solve is negative 16x squared plus 20x plus 6 equals 0. We can solve this equation using the quadratic formula. From the equation, we can see that a equals negative 16, b is 20, and c is 6. So substituting these values into the formula gives us the expression here.
Simplifying the denominator is easy. 2 times negative 16 is negative 32. When we simplify the numerator, it's more complicated because it involves a plus/minus symbol, square roots, and other operations.
We'll start to simplify underneath the square root and begin with our exponent. 20 squared is 400. And then multiplication, 4 times negative 16 times 6 is a negative 384. We then subtract 400 minus negative 384, which is 784.
Finally, we take the square root of 784, which is 28. So now, our expression becomes negative 20 plus or minus 28 over negative 32. We can now separate our solution into two fractions-- first fraction being negative 20 over negative 32 plus or minus 28 over negative 32. The plus and minus symbol gives us our two solutions. The first solution is negative 20 over negative 32 plus 28 over negative 32.
And the second solution is negative 20 over negative 32 minus 28 over negative 32. Simplifying these two fractions gives us 20 over 32 minus 28 over 32 and 20 over 32 plus 28 over 32. Adding and subtracting gives us negative 8 over 32 and 48 over 32. Dividing each of these fractions gives us negative 0.25 and 1.5. So our solutions to this quadratic equation are x equals negative 0.25 and x equals 1.5.
Going back to our original problem, we want to find the time it takes to reach the ground. So our solution of x equals negative 0.25 can be disregarded because it doesn't make sense in the context of the problem. Therefore, it takes 1.5 seconds for the shot to hit the ground.
Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. Factoring or variable isolation may be used to solve some but not all quadratic equations. However, the quadratic formula can be used to find solutions to all quadratic equations even when factoring or variable isolation is difficult or impossible. Before using the quadratic formula, the equation must be equal to 0 in order to determine the correct values of a, b, and c to use in the formula.
So I hope that these important points and examples helped you understand a little bit more about the quadratic formula. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.