Vieta's formulas

Imagine you have a bunch of numbers that you want to multiply together to get some answer. For example, let's say you want to multiply three numbers, like 2, 3, and 4. You can write this like:

2 x 3 x 4 = 24

Now, what if you wanted to know the sum of those same three numbers? Well, you could add them up like this:

2 + 3 + 4 = 9

Vieta's formulas are a way to use these ideas of multiplication and addition to help solve mathematical problems. Specifically, they deal with what are called polynomial equations. These are equations where you have a bunch of numbers that are being added and multiplied together, like this:

3x^2 + 2x + 1 = 0

Vieta's formulas give us a way to find the solutions to this kind of equation. The solutions are the values of x that make the equation true. So, in this case, we want to find the values of x that make 3x^2 + 2x + 1 equal to zero.

To use Vieta's formulas, we start by looking at the coefficients of the polynomial. The coefficients are the numbers that are being multiplied by the variables (in this case, x). So, in our example, the coefficients are 3, 2, and 1.

First, Vieta's formulas tell us that the sum of the solutions to the equation is equal to the opposite of the coefficient of the x-term (the term with x in it) divided by the coefficient of the highest power of x. In our example, this means that:

sum of solutions = -2/3

So, we know that the sum of the solutions is equal to negative two-thirds.

Next, Vieta's formulas tell us that the product of the solutions is equal to the constant term (the number at the end of the polynomial, in this case 1) divided by the coefficient of the highest power of x. In our example, this means that:

product of solutions = 1/3

So, we know that the product of the solutions is equal to one-third.

Using these two pieces of information, we can then find the actual solutions to the equation. In this case, we can see that there are two solutions (since it's a quadratic equation). Let's call them x1 and x2. We know that:

x1 + x2 = -2/3

x1 x x2 = 1/3

We can use these two equations to solve for x1 and x2. I won't go into the details of how to do this, since it gets a little more complicated, but the point is that Vieta's formulas give us a way to solve for the solutions to polynomial equations using just the coefficients of the polynomial.