# Graphical solution of quadratic equations

What is the largest number of times that a parabola can cross the $x\text{-axis}$?

## Graphical solution of quadratic equations

Too steep, huh? Okay, let's change the slope a bit... Still hard? Yeah, you should be glad it's not a quadratic function. 'Cause those get steeper and steeper, the longer you walk. Look here: Y equals X squared.

That little two there, that's what makes it a quadratic function. Let's write down a value table, like this, and it'll be clearer. When X is zero, then Y is zero-squared; zero! X is one, Y is one squared: one. X is two, Y is two squared: four!

And so on. Look: The graph starts off horizontally, but it quickly gets steeper and steeper. Turn around and watch, and you'll see what happens if X is negative. It goes upward there too! It's the X-squared that gives it this shape.

If you see a function with X to the second power, then you know it doesn't describe a straight line, but an arc -- a parabola. Then it is a quadratic function. Let's take another one, a bit harder. Here's a quadratic function with three terms, An X-squared term an X-term and a constant. Let's take them one by one.

First, a value table... ... with columns for X and Y. Choose a couple of values for X, place them in the equation, and calculate Y. 'Y' equals 'X squared' minus 'two X' minus 'three'. If 'X' is zero, then 'Y' equals 'negative three'... 'X' equals 'one' gives 'Y' equals 'minus four'. 'X' equals 'two' gives 'Y' equals 'minus three'. 'X' equals 'three' gives 'Y' equals 'zero'. And what is 'Y' if 'X' equals 'negative one'?

Zero! Now we have calculated a few points and marked them. Now, we fill out a nice and smooth graph, through the points. This can be used to solve equations too, and that's useful, when you run into an equation where one of the terms is squared. Set Y to zero, and get this equation: 'Zero' equals 'X squared' minus 'two X' minus 'three'.

Since we already have the graph, we won't have to calculate at all to solve this equation. We have set Y to 'zero', so all we have to do is to find a point on the graph where Y equals 'zero'. And Y equals zero along the entire X axis -- so, we will find the solution to the equation, where the graph intersects the X axis! And that is... there!

Y equals zero where X equals ... three! But hang on a moment! There's another intersect! There!

Y is zero where X is -- minus one! What does this mean? Can there be more than one solution to an equation? Test for yourself! Pause the film, and set X to three in the equation.

Then test setting it to negative one. Both solutions work! A quadratic equation can have two solutions! Pretty neat huh? Write a function that has an 'X squared' term, and you'll have a quadratic function.

Write up a value table, so you can draw the graph of the quadratic function. The graph of a quadratic function is bent into a parabola. Set Y to zero, and you have a quadratic equation that you can solve graphically by finding where the graph intersects the X axis. A quadratic equation can have two solutions. Quadratic equations appear all over mathematics, physics, economics... ...