Brenda King

with a graphical structure

Introduction:

The quadratic formula, shown below, is used to find the solution to quadratic equation in the form ax2 + bx + c = 0.

The pattern of roots or solutions to quadratic equations can be found by keeping two of the parameters constant and letting the third parameter vary.

For example, if we set a=1, c=1 and let b vary the equation becomes 1x2 + bx + 1 = y.

If we go further and let y=0, the equation can be solved depending on the assignment of parameter b.

Solving for b in the equation 1x2 + bx + 1 = 0, we get .

How can this graph be related to the quadratic formula?

Exploration:

Taking the derivative of 1x2 + bx + 1 = 0 with respect to x gives the line 2x + b = 0. Rewriting this becomes .

This line will go through the turning points of the graph as shown below and represents part of the quadratic formula.

Diagram 1

Specific Case:

If we consider a particular value of b, say b=4, then the x-values become roots of 1x2 + 4x + 1 = 0

The graph the line on the xb plane it would look like this:

Diagram 2

The line b=4 intersects the curve  at the points  and (see green dots in diagram 2).

The  midpoint, m, would be located at (see red dot in diagram 2).  This point also lies on the line 2x + 4 = 0.

The distance between the midpoint and the curve is .

In General:

The point marked in red, m, is a midpoint between the edges of the graph  as shown in the specific case when b = 4.

Solving the line 2x+b=0 for b gives a general form  for the midpoint.  Another way to find the midpoint would be to work with a general point.

To find a general point on the curve, 1x2 + bx + 1 = 0, it is easier to use the vertex form of the equation, .

Derivation:

The general point on the curve of 1x2 + bx + 1 = 0 is

By using this point, the midpoint formula,  and distance formula,  the following results can be found.

Midpoint  and distance between the midpoint and the curve = .

The quadratic formula, at a particular value of b, is a point on the curve 1x2 + bx + 1 = 0.

This point is the root and can be found on the line 2x+b  distance .

When b>2 there would be two negative real roots.

When b=2 or b=-2, there would be one real root.

When –2 < b < 2, no real roots .

When b < -2 . there would be two positive real roots.