Take a look at the applet: Sine Functions
The general form of the quadratic formula is .
It is not immediately apparent how this function rule could be derived by transformation of the basic power function . That makes it difficult to find the vertex and x-intersects of the corresponding parabola.
Using a method called completing the square allows you to convert the function to the form: where are the coordinated of the vertex of the graph.
To do this conversion you use the following property:
Use the applet to check that is the same function as .
It is obviously very useful if by completing the square you can convert to a form that allows you to immediately see the vertex and the axis of symmetry...
A long time ago, mathematicians derived the so-called quadratic formula.
This formula allows you to solve and thereby find the zeros of the quadratic equation. The general solution is:
Below you see a proof of the quadratic formula. This means you can show that the formula is always valid. To do so you need to solve in general terms by completing the square.
Assume that (otherwise it would not be a quadratic equation!). You now divide by on both sides. This gives you:
Completing the square results in:
Taking the square root:
And now a few rearrangements:
The quadratic formula has been derived..
The expression in the root is called the discriminant of the quadratic equation. Since only the root of a positive number or is itself a real number, it is the value of the discriminant that determines the number of solutions of the equation:
and there are two solutions;
and there is one solution (or the same solution twice);
and there are no real solutions;