Solutions or Roots of Quadratic Equations

Consider the quadratic equation

displaymath22

A real number x will be called a solution or a root if it satisfies the equation, meaning tex2html_wrap_inline26 . It is easy to see that the roots are exactly the x-intercepts of the quadratic function tex2html_wrap_inline28 , that is the intersection between the graph of the quadratic function with the x-axis.

a<0
a>0

Example 1: Find the roots of the equation

displaymath30

Solution. This equation is equivalent to

displaymath32

Since 1 has two square-roots tex2html_wrap_inline34 , the solutions for this equation are

displaymath36

Example 2: Find the roots of the equation

displaymath38

Solution. This example is somehow trickier than the previous one but we will see how to work it out in the general case. First note that we have

displaymath40

Therefore the equation is equivalent to

displaymath42

which is the same as

displaymath44

Since 3 has two square-roots tex2html_wrap_inline46 , we get

displaymath48

which give the solutions to the equation

displaymath50

We may then wonder whether any quadratic equation may be reduced to the simplest ones described in the previous examples. The answer is somehow more complicated but it was known for a very longtime (to the Babylonians about 2000 B.C. ). Their idea was based mainly on completing the square which we did in solving the second example.

[Algebra] [Complex Variables]
[Geometry] [Trigonometry ]
[Calculus] [Differential Equations] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2014 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour