Power functions > The quadratic formula
12345The quadratic formula

Theory

Take a look at the applet: Sine Functions

The general form of the quadratic formula is f ( x ) = a x 2 + b x + c .
It is not immediately apparent how this function rule could be derived by transformation of the basic power function y = x 2 . 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 f to the form: f ( x ) = a ( x p ) 2 + q where ( p , q ) are the coordinated of the vertex of the graph.
To do this conversion you use the following property:

x 2 + 2 k x = ( x + k ) 2 k 2

Use the applet to check that f ( x ) = 2 x 2 4 x is the same function as g ( x ) = 2 ( x 1 ) 2 2 .

It is obviously very useful if by completing the square you can convert f ( x ) = a x 2 + b x + c 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 a x 2 + b x + c = 0 and thereby find the zeros of the quadratic equation. The general solution is:
x = - b + b 2 - 4 a c 2 a x = - b - b 2 - 4 a c 2 a

> proof

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    a x 2 + b x + c = 0 in general terms by completing the square.

Assume that a 0 (otherwise it would not be a quadratic equation!). You now divide by a on both sides. This gives you:

x 2 + b a x + c a = 0

Completing the square results in:

( x + b 2 a ) 2 ( b 2 a ) 2 + c a = 0 and ( x + b 2 a ) 2 = ( b 2 a ) 2 c a = b 2 4 a c 4 a 2

Taking the square root:

x + b 2 a = ± b 2 4 a c 4 a 2

And now a few rearrangements:

x = - b 2 a ± b 2 4 a c 4 a 2 = - b 2 a ± b 2 4 a c 2 a = - b ± b 2 4 a c 2 a

The quadratic formula has been derived..

The expression D = b 2 4 a c in the root is called the discriminant of the quadratic equation. Since only the root of a positive number or 0 is itself a real number, it is the value of the discriminant that determines the number of solutions of the equation:

  • D > 0 and there are two solutions;

  • D = 0 and there is one solution (or the same solution twice);

  • D < 0 and there are no real solutions;

previous | next