Solutions or Roots of Quadratic Equations

Consider the quadratic equation


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.


Example 1: Find the roots of the equation


Solution. This equation is equivalent to


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


Example 2: Find the roots of the equation


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


Therefore the equation is equivalent to


which is the same as


Since 3 has two square-roots tex2html_wrap_inline46 , we get


which give the solutions to the equation


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]
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[Calculus] [Differential Equations] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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