In this deck, we'll explore a powerful technique called iteration. Sometimes, equations are too difficult to solve exactly. Iteration gives us a way to find an approximate solution by repeating a simple calculation over and over, getting closer to the answer with each step.

Think of it like a 'hotter or colder' game. You make a guess, you're told how to make a better guess, and you repeat the process until you zero in on the target. In maths, we use a special formula to guide our guesses.

Before we start iterating, we often need to prove that a solution even exists within a certain range. We do this by looking for a change of sign.

Imagine a continuous function's graph. If the graph is below the x-axis at one point, giving a negative value, and above the x-axis at another point, giving a positive value, it must have crossed the x-axis somewhere in between. That crossing point is our solution.

Let's test the equation x squared minus 7 equals zero, between x equals 2 and x equals 3. When x is 2, x squared is 4, and 4 minus 7 is negative 3. When x is 3, x squared is 9, and 9 minus 7 is positive 2.