Brenda King

Presenting solutions to Quadratic
equations

with a graphical structure

__Introduction:__

The quadratic formula,
shown below, is used to find the solution to quadratic equation in the form ax^{2
}+ 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 1x^{2 }+ 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 1x^{2 }+ bx + 1 = 0, we get _{}.

How can this graph
be related to the quadratic formula?

__Exploration:__

Taking the
derivative of 1x^{2 }+ 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 1x^{2 }+ 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, 1x^{2 }+ bx + 1 = 0, it
is easier to use the vertex form of the equation, _{}.

Derivation:

_{}

_{}

_{}

_{}

The general point
on the curve of 1x^{2 }+ 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 1x^{2 }+ 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.