Tarski's high school algebra problem

Okay kiddo, let's start with what algebra is. You know how sometimes we use letters instead of numbers to represent something we don't know? Well, that's algebra! We use letters called variables to solve math problems.

Now, Tarski's high school algebra problem is a really cool math problem that was created by a guy named Alfred Tarski. He was a very smart mathematician who came up with this tricky question.

The problem goes like this: There are two numbers, x and y. The sum of the numbers is 10 and the product of the numbers is 21. What are the two numbers?

It seems like a simple problem, right? But it can be tricky to figure out the answer. So let's break it down step by step.

First, we know that the sum of the two numbers is 10. That means if we add x and y together, the answer will be 10. We can write this as:

x + y = 10

Next, we know that the product of the two numbers is 21. That means if we multiply x and y together, the answer will be 21. We can write this as:

xy = 21

Now, we have two equations:

x + y = 10
xy = 21

To solve for x and y, we can use a method called substitution. We take one of the equations and solve it for one variable, then substitute that expression in for that variable in the other equation. Here's how we do it:

Let's solve the first equation for one of the variables. We can solve for y by subtracting x from both sides:

y = 10 - x

Now, we can substitute this expression for y into the second equation:

x(10-x) = 21

We can simplify this by combining like terms:

10x - x^2 = 21

Rearranging the terms:

x^2 - 10x + 21 = 0

Now we have a quadratic equation, which means it has an x^2 term. We can solve for x using something called the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

But let's not worry too much about that right now. The important thing is that we can solve for x and y by using this equation.

When we solve for x using the quadratic formula, we get two answers:

x = 3 or x = 7

Now we can substitute these values back into the equation y = 10 - x to find the value of y:

If x = 3, then y = 10 - 3 = 7

If x = 7, then y = 10 - 7 = 3

So the two numbers are either 3 and 7 or 7 and 3.

And that, my little friend, is how we solve Tarski's high school algebra problem!