Advertisement.

EnchantedLearning.com is a user-supported site.
As a bonus, site members have access to a banner-ad-free version of the site, with print-friendly pages.
Click here to learn more.

ad
(Already a member? Click here.)

Our subscribers' grade-level estimate for this page: 5th - 6th
quadratic equation
More Math Activities
EnchantedLearning.com
Algebra:
Quadratic Equations
quadratic formula
Quadratic Equation: ax2 - c = 0

Printable Worksheets


A quadratic equation is an equation that has a second-degree term and no higher terms. A second-degree term is a variable raised to the second power, like x2. When you graph a quadratic equation, you get a parabola, and the solutions to the quadratic equation represent where the parabola crosses the x-axis.

A quadratic equation can be written in the form:

quadratic equation,
where a, b, and c are numbers (a ≠0), and x is the variable. x is a solution (or a root) if it satisfies the equation ax2 + bx + c = 0.

Some examples of quadratic equations include:
3x2 + 9x - 2 = 0 6x2 + 11x = 7 4x2 = 13


Solving a Quadratic Formula:
Some quadratic equations can be solved easily by factoring. Some simple-to-solve quadratic equations are:

x2 - 1 = 0

(x + 1)(x - 1) = 0

x = ±1

x2 - 5x + 6 = 0

(x - 2)(x - 3) = 0

x = 2, 3

Most second-degree equations are more difficult to solve, and cannot be solved by simple factoring. The quadratic formula is a general way of solving any quadratic equation:

quadratic formula

This formula gives two solutions, although the two solutions may be the same number. (When solving any polynomial equation of degree n, there are at most n solutions to that equation.)


Deriving the Quadratic Formula:
The quadratic formula is obtained by solving the general quadratic equation. This is one way to derive the quadratic formula:

Deriving  Quadratic
Divide each side of the equation by a.
Deriving Quadratic
Subtract c/a from each side of the equation.
Deriving Quadratic
Add (b/2a)2 to each side of the equation (to complete the square).
Deriving Quadratic
Find a common denominator for the right side of the equation.
Deriving Quadratic
Take the square root of each side of the equation.

Deriving Quadratic
Add b/2a to each side of the equation.

Deriving Quadratic
The plus-or-minus sign shows that there are two possible solutions.


parabolas
  • The green parabola has 2 x-intercepts. Its corresponding quadratic equation has 2 distinct solutions (x=1 and x=4).
  • The yellow parabola has 1 x-intercept. Its corresponding quadratic equation has 1 solution (x=-3).
  • The purple parabola has no x-intercepts. Its corresponding quadratic equation has no solutions.
You can try to solve any quadratic equation by using the quadratic formula (or by solving it algebraically, similar to the way we just derived the quadratic formula above).

Every quadratic equation has at most two solutions, but for some equations, the two solutions are the same number, and for others, there is no solution on the number line (because it would involve the square root of a negative number).

One way to understand this visually is to realize that when you graph a quadratic equation (a second-degree equation), you get a parabola. When you set the quadratic equation equal to zero, this represents the points where the parabola hits the x-axis (the x-intercepts, where y=0).


Introduction to Algebra - Printable Worksheets
Quadratic

Quadratic Equations Worksheets (with no linear term)

Solve simple (pure) quadratic equations (no linear terms).


Enchanted Learning®
Over 35,000 Web Pages
Sample Pages for Prospective Subscribers, or click below

Overview of Site
What's New
Enchanted Learning Home
Monthly Activity Calendar
Books to Print
Site Index

K-3
Crafts
K-3 Themes
Little Explorers
Picture dictionary
PreK/K Activities
Rebus Rhymes
Stories
Writing
Cloze Activities
Essay Topics
Newspaper
Writing Activities
Parts of Speech

Fiction
The Test of Time
iPhone app
TapQuiz Maps - free iPhone Geography Game

Biology
Animal Printouts
Biology Label Printouts
Biomes
Birds
Butterflies
Dinosaurs
Food Chain
Human Anatomy
Mammals
Plants
Rainforests
Sharks
Whales
Physical Sciences: K-12
Astronomy
The Earth
Geology
Hurricanes
Landforms
Oceans
Tsunami
Volcano
Languages
Dutch
French
German
Italian
Japanese (Romaji)
Portuguese
Spanish
Swedish
Geography/History
Explorers
Flags
Geography
Inventors
US History

Other Topics
Art and Artists
Calendars
College Finder
Crafts
Graphic Organizers
Label Me! Printouts
Math
Music
Word Wheels

Click to read our Privacy Policy

E-mail


Enchanted Learning Search

Search the Enchanted Learning website for:



Advertisement.



Advertisement.



Advertisement.


Copyright ©2005 EnchantedLearning.com ------ How to cite a web page