Consider the quadratic equation

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

** Solution.** This equation is equivalent to

Since 1 has two square-roots , 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 , 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.

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