Processing Math: Done

From Förberedande kurs i matematik 1

Let the point on the x-axis have coordinates x0 , where x is an unknown number. Then, using the distance formula, the distance from the point x0  to 33  and 51  is given by

x−32+0−32  and x−52+0−12 

Because these two distances should be the same, we get the equation

x−32+9=x−52+1 

or, if we take the square, so as to get rid of the square root sign,

x−32+9=x−52+1 

Now, expand the squares and collect together all the terms onto one side:

x2−6x+9+9=x2−10x+25+14x−8=0

This gives that x=2, i.e. the point on the x-axis is 20 .

As a final step, we check that we have calculated correctly and that the distances really are the same. The distance between 20  and 33  is

3−22+3−02=12+32=1+9=10 

and the distance between 20  and 51  is

5−22+1−02=32+12=9+1=10 

NOTE: Although we squared our root equation, it is not in fact necessary to test the solution for that reason, because the expressions under the root signs are sums of squares which are never negative and therefore cannot give rise to so-called false roots.