Quadratic Equation
Pronunciation: /kwɒˈdræ tɪk ɪˈkweɪ ʃən/ ?
 Figure 1: Graph of the quadratic equation f(x) = x^{2}+2x. 
 Manipulative 1: Quadratic equation. Created with GeoGebra. 
A quadratic equation is an
equation
of a
polynomial
of degree
two. When graphed, a quadratic equation makes a
parabola
with a horizontal directrix. Figure 1 shows a graph of the quadratic equation
f(x) = x^{2}+2x.
The standard form of a quadratic equation is
f(x) = ax^{2} + bx + c where a, b and c
are constant coefficients and a≠0. The equation in figure 1 placed
in standard form is
f(x) = x^{2} + 2x + 0. Manipulative 1 shows the graph of a quadratic
equation using the standard from. Click on the red points on the sliders in manipulative
1 to change the figure. Click on the reset
button to change the figure back to its original configuration.
Discovery
 Move the slider for a until a=0. What is the result?
Why is the condition a≠0 placed on the quadratic equation?
 What changes in the graph when a is changed?
 What changes in the graph when b is changed?
 What changes in the graph when c is changed?
Examples of Quadratic Equations 
Quadratic equations  Reason 
y = 3x^{2}  2x  4  Function is a polynomial of degree 2. 
y = x^{2}  3  Function is a polynomial of degree 2. 
g = b^{2}  Function is a polynomial of degree 2. 
y = sin(π/2)x^{2} + 4  Function is a polynomial of degree 2. Since the argument of the sine function is a constant, the function itself is a constant and can be treated like any other coefficient. 
r = (t1)(t+3)  The right side of the equation can be expanded using the distributive property of multiplication over addition and subtraction to yield r = t^{2} + 2t  3, which is a polynomial of degree 2. 
Examples of Equations That Are Not Quadratic 
Nonquadratic equations  Reason 
y = 3x^{3} + 2x^{2}  x + 3  Function is a polynomial of degree 3, so it is not quadratic. 
y = x^{2}  sin(x) + 3  Since sin(x) is in the equation, it is not a polynomial. 
w = 3m  4  Function is a polynomial of degree 1, so it is not quadratic. 
y = log(2x^{2}  x + 3)  Since the equation contains a logarithm with a variable argument, it is not a polynomial. 
y = x(x2)(x+1)  The right side of the equation can be expanded to y = x^{3}  x^{2}  2x, which is a polynomial of degree 3. 
Table 1 

Discriminant of a Quadratic Equation
 Figure 2: Discriminants of quadratic equations. 

The discriminant of a quadratic equation is used to determine if a quadratic
equation has real or complex roots. The expression for the discriminant is
.
If the
discriminant is positive, the quadratic equation has two real roots. If the
discriminant is zero, the quadratic equation has one real root. If the
discriminant is negative, the quadratic equation has two complex roots.

For a quadratic equation in the form
,
the solution can be found using the
quadratic formula
.
A quadratic equation can have 2 real roots, 1 real root, or 2 complex roots
(see
discriminant).
Example 1: Two real roots.
  Figure 3: Graph of x^{2}+2x3. 

Example 2: One real root.
  Figure 4: Graph of x^{2}+4x+4. 

Example 3: Two complex roots.
  Figure 5: Graph of x^{2}+4=0. 

Forms of Quadratic Equations
 Manipulative 2: Parabolic Function  Normal Form. Manipulative created with GeoGebra. 

The standard form a parabolic equation is
y = ax^{2} + bx + c. In manipulative 2, click on the points on the
red sliders and drag them to change the figure.
Discovery
 Click on the point for the slider labeled 'a' and drag it. What changes if 'a'
is negative, zero or positive?
 Click on the point for the slider labeled 'b' and drag it. What changes as 'b'
changes?
 Click on the point for the slider labeled 'c' and drag it. What changes as 'c'
changes?

 Manipulative 3: Parabolic Function  Xintercept Form. Manipulative created with GeoGebra. 

Xintercept
form of a parabolic equation is y = a(xx_{0})(xx_{1})
where x_{0} is one xintercept of the quadratic equation,
x_{1} is the other xintercept, and a indicates how
steep the sides of the quadratic equation are. If x_{0} = x_{1},
the quadratic equation intercepts the xaxis only once. Not all quadratic equations can be described
using the xintercept form.
Manipulative 3 shows a quadratic equation with elements of the xintercept form. Click on
the xintercepts labeled xi_{0} and xi_{1}
and drag them to change the figure. Click on the point labeled a on
the slider and drag it to change the figure.
Discovery
 Click on the blue points labeled xi_{0} and
xi_{1} and drag them. How do they change the figure?
 Click on the red point on the slider labeled a and drag it.
How does it change the figure?
 Can you use manipulative 3 to show a quadratic equation that does not intercept the
xaxis? Why or why not?

 Manipulative 4: Parabolic Function in Vertex Form. Manipulative created with GeoGebra. 

The vertex form of a parabolic equation is
yy_{0} = a(xx_{0})^{2}. The vertex of
the quadratic equation is
at the point (x_{0},y_{0}). a shows how
steep the sides of the quadratic equation are. Click on the points on the sliders in
manipulative 4 and drag them to change the figure.
Discovery
 Click on the sliders for x_{0} and y_{0}
and drag them. What changes about the quadratic equation?
 Click on the slider for a and drag it. What changes about the quadratic equation?
 If yy_{0} = a(xx_{0})^{2} is the equation
of the quadratic equation, what is the equation of the line of symmetry?

 Manipulative 5: Parabolic Function in Conic Form. Manipulative created with GeoGebra. 
The conic section form of a parabolic equation is
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 where
B^{2} = 4AC and either A ≠ 0 or C ≠ 0.
This form allows one to draw quadratic equations where the directrix is not parallel to the
xaxis or yaxis. The sliders in manipulative 5 allow you to change the figure.
Note that for some values of A through F, the quadratic equation will
be out of the display area of the graph.
Discovery
 A criteria for the conic section form of a parabolic is A ≠ 0
or C ≠ 0. What happens if both A and C are
0?
 What direction does the parabola open if C = 0? Make sure to change
A to both positive and negative to get the full picture.
 What direction does the parabola open if A = 0? Make sure to change
C to both positive and negative to get the full picture.

Graphing a Quadratic Equation
The graph of a quadratic equation can be drawn using
the definition of a parabola: All points in a plane
equidistant
from a line, called the directrix, and a point, called the
focus, which is not on the line. Every quadratic equation
also has a
vertex
and a
line of symmetry.
Discovery
In manipulative 6, click on the focus and the red point on the directrix and drag them
change the figure.
 How is the quadratic equation different if the focus and directrix are close
together as opposed to far apart?
 How does the quadratic equation change if the focus is moved
to the left or to the right?
 Click on the focus (purple point) and drag it until the focus is on the directrix.
What happens to the quadratic equation? When this happens, mathematicians say that the line
is a degenerate quadratic equation.
Parts of a Parabola
 Manipulative 6: Parts of a parabola Manipulative created with GeoGebra. 
The vertex of a quadratic equation is at the inflection point of the quadratic equation. The inflection
point is the point where the parabolic curve changes direction. In manipulative 6,
the vertex is blue. The vertex can not be dragged in this manipulative, because the
vertex is dependent upon the focus and directrix. Click on
the check boxes in manipulative 2 to see the parts of a quadratic equation.
The distance from the focus to any point on the quadratic equation is the same as the distance
from that point to the directrix. Click on the blue point on the quadratic equation and drag it
to demonstrate this property.

More Information
Cite this article as:
Quadratic Equation. 20090122. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/q/quadraticequation.html.
Translations
Image Credits
Revision History
20090122: Added figure to section on discriminants (
McAdams, David.)
20081210: Added section on discriminant, solutions to a quadratic equation (
McAdams, David.)
20081019: Initial version (
McAdams, David.)