Do all rational equations have a single solution? Why is that so?
Whenever you solve a rational equation you must make sure the result obtained for
an answer does not allow the denominator of one of the rational expressions to assume a value of ZERO, [ as division by zero is undefined and therefore prohibited.
For example if we have
2x/(x-3) =(x2 -9x)/ x
when we multiply out by x(x-3) we get
2x(x) = (x2 -9x)(x-3)
so 2x2 = x(x-9)(x-3)
2x2 = x(x2 - 12x + 27)
= x3 - 12x2 + 27x
0 = x3 - 14x2 + 27x
0 = x(x2 - 14x + 27)
so solutions are 0 and 7 + v22 and 7 -v22
but 0 makes right hand side expression have zero in denominator so it is not
We actually have to look at all obtained solutions to be sure they ae not extraneous.
Suppose we had obtained a 3 for a solution. That would make the left side denominator equal zero and we would have to dismiss that, if 3 was obtained.
The two irrational solutions we have obtained are genuine solutions as neither introduces a zero to a denominator
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