Quadratic Equations

Definition

ax2 + bx + c = 0

a, b, c are constants (generally integers)

Roots

Synonyms: Solutions or Zeros 

Consider the graph of quadratic equations. The quadratic equation looks like ax2 + bx + c = 0, but if we take the quadratic expression on the left and set it equal to y, we will have a function:

y = ax2 + bx + c

When we graph y vs. x, we find that we get a curve called a parabola. The specific values of a, b, and c control where the curve is relative to the origin (left, right, up, or down), and how rapidly it spreads out. Also, if a is negative then the parabola will be upside-down. What does this have to do with finding the solutions to our original quadratic equation? Well, whenever y = 0 then the equation y = ax2 + bx + c is the same as our original equation.

Graphically, y is zero whenever the curve crosses the x-axis. Thus, the solutions to the original quadratic equation (ax2 + bx + c = 0) are the values of x where the function (y = ax2 + bx + c) crosses the x-axis. From the figures below, you can see that it can cross the x-axis once, twice, or not at all.

 

 Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

 

    1. Move all the terms to one side, so that it is equal to zero
    2. Set the resulting expression equal to y (in place of zero)
    3. Enter the function into your calculator and graph it
    4. Look for places where the graph crosses the x-axis

 

Your graphing calculator most likely has a function that will automatically find these intercepts and give you the x-values with great precision. Of course, no matter how many decimal places you have it is still just an approximation of the exact solution. In real life, though, a close approximation is often good enough.