Quadratic Equation

Pronunciation: /kwɒˈdræ tɪk ɪˈkweɪ ʃən/ ?

Graph of the quadratic equation y=x^2+2x, which is a parabola with vertex at (-1,-1).
Figure 1: Graph of the quadratic equation f(x) = x2+2x.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
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) = x2+2x.

The standard form of a quadratic equation is f(x) = ax2 + 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) = x2 + 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 Geogebra reset button button to change the figure back to its original configuration.

Discovery

  1. Move the slider for a until a=0. What is the result? Why is the condition a≠0 placed on the quadratic equation?
  2. What changes in the graph when a is changed?
  3. What changes in the graph when b is changed?
  4. What changes in the graph when c is changed?

Examples of Quadratic Equations

Quadratic equationsReason
y = 3x2 - 2x - 4Function is a polynomial of degree 2.
y = -x2 - 3Function is a polynomial of degree 2.
g = b2Function is a polynomial of degree 2.
y = sin(π/2)x2 + 4Function 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 = (t-1)(t+3)The right side of the equation can be expanded using the distributive property of multiplication over addition and subtraction to yield r = t2 + 2t - 3, which is a polynomial of degree 2.

Examples of Equations That Are Not Quadratic

Non-quadratic equationsReason
y = 3x3 + 2x2 - x + 3Function is a polynomial of degree 3, so it is not quadratic.
y = x2 - sin(x) + 3Since sin(x) is in the equation, it is not a polynomial.
w = 3m - 4Function is a polynomial of degree 1, so it is not quadratic.
y = log(2x2 - x + 3)Since the equation contains a logarithm with a variable argument, it is not a polynomial.
y = x(x-2)(x+1)The right side of the equation can be expanded to y = x3 - x2 - 2x, which is a polynomial of degree 3.
Table 1

Discriminant of a Quadratic Equation

Three quadratic equations showing the number of roots and the discriminants.
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
b^2-4ac.
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.

Solutions to a Quadratic Equation

For a quadratic equation in the form ax^2+bx+c=0, the solution can be found using the quadratic formula
x=(-b+-square root(b^2-4ac))/(2a).
A quadratic equation can have 2 real roots, 1 real root, or 2 complex roots (see discriminant).

Example 1: Two real roots.

x^2+2x-3=0 implies x=(-2+-square root(2^2-4*1*(-3)))/(2*1) implies x=(-2+-square root(4+12))/2 implies x=(-2+-square root(16))/2 implies x=(-2+-4)/2 implies x=-1+-2 implies x=-3 or x=1.
Graph of f(x)=x^2+2x-3.
Figure 3: Graph of x2+2x-3.

Example 2: One real root.

x^2+4x+4=0 implies x=(-4+-square root(4^2-4*1*4))/(2*1) implies x=(-4+-square root(16-16))/2 implies x=(-4+-square root(0))/2 implies x=-4/2 implies x=-2.
Graph of f(x)=x^2+4x+4.
Figure 4: Graph of x2+4x+4.

Example 3: Two complex roots.

x^2+4=0 implies x=(-0+-square root(0^2-4*1*4))/(2*1) implies x=(+-square root(0-16))/2 implies x=(+-square root(-16))/2 implies x=+-4i/2 implies x=2i or x=-2i.
Graph of f(x)=x^2+4.
Figure 5: Graph of x2+4=0.

Forms of Quadratic Equations

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 2: Parabolic Function - Normal Form. Manipulative created with GeoGebra.

Standard Form

The standard form a parabolic equation is y = ax2 + bx + c. In manipulative 2, click on the points on the red sliders and drag them to change the figure.

Discovery

  1. Click on the point for the slider labeled 'a' and drag it. What changes if 'a' is negative, zero or positive?
  2. Click on the point for the slider labeled 'b' and drag it. What changes as 'b' changes?
  3. Click on the point for the slider labeled 'c' and drag it. What changes as 'c' changes?

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 3: Parabolic Function - X-intercept Form. Manipulative created with GeoGebra.

X-intercept Form

X-intercept form of a parabolic equation is y = a(x-x0)(x-x1) where x0 is one x-intercept of the quadratic equation, x1 is the other x-intercept, and a indicates how steep the sides of the quadratic equation are. If x0 = x1, the quadratic equation intercepts the x-axis only once. Not all quadratic equations can be described using the x-intercept form.

Manipulative 3 shows a quadratic equation with elements of the x-intercept form. Click on the x-intercepts labeled xi0 and xi1 and drag them to change the figure. Click on the point labeled a on the slider and drag it to change the figure.

Discovery

  1. Click on the blue points labeled xi0 and xi1 and drag them. How do they change the figure?
  2. Click on the red point on the slider labeled a and drag it. How does it change the figure?
  3. Can you use manipulative 3 to show a quadratic equation that does not intercept the x-axis? Why or why not?

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 4: Parabolic Function in Vertex Form. Manipulative created with GeoGebra.

Vertex Form

The vertex form of a parabolic equation is y-y0 = a(x-x0)2. The vertex of the quadratic equation is at the point (x0,y0). 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

  1. Click on the sliders for x0 and y0 and drag them. What changes about the quadratic equation?
  2. Click on the slider for a and drag it. What changes about the quadratic equation?
  3. If y-y0 = a(x-x0)2 is the equation of the quadratic equation, what is the equation of the line of symmetry?

Conic Section Form

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 5: Parabolic Function in Conic Form. Manipulative created with GeoGebra.

The conic section form of a parabolic equation is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where B2 = 4AC and either A ≠ 0 or C ≠ 0. This form allows one to draw quadratic equations where the directrix is not parallel to the x-axis or y-axis. 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

  1. 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?
  2. What direction does the parabola open if C = 0? Make sure to change A to both positive and negative to get the full picture.
  3. 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.

  1. How is the quadratic equation different if the focus and directrix are close together as opposed to far apart?
  2. How does the quadratic equation change if the focus is moved to the left or to the right?
  3. 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

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
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. 2009-01-22. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/q/quadraticequation.html.

Translations

Image Credits

Revision History


2009-01-22: Added figure to section on discriminants (McAdams, David.)
2008-12-10: Added section on discriminant, solutions to a quadratic equation (McAdams, David.)
2008-10-19: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License