Could somebody guide me about this which is about the relation between the roots and the coefficients?
For what values of m will the equation have
a) one root the reciprocal of the other,
b) one root equal zero
c) equal roots
To do any of those we have to first get
in the form
----------------
a) one root the reciprocal of the other
Suppose one root is r and the other is , then
the quadratic equation with this property and leading
coefficient 1 is found this way:
Therefore this must be the same equation as
So we equate like parts:
or simplifying:
To check that, we substitute
for :
,
,
So both roots are equal and -1, but -1 is the
reciprocal of -1.
So the answer to (a) is
-------------
b) one root equal zero
Suppose one root is 0 and the other is r, then
the quadratic equation with this property and leading
coefficient 1 is found this way:
Therefore this must be the same equation as
So we equate like parts:
Simplifying:
So we see that m must be -7.
To check that, we substitute
for :
,
One root is 0, so we are correct.
So the answer to (b) is
-------------
c) equal roots
Suppose one root is r and the other is also r, then
the quadratic equation with this property and leading
coefficient 1 is found this way:
Therefore this must be the same equation as
So we equate like parts:
Simplifying:
Solve both equations for m
Equate the right sides since both equal m:
,
,
Substituting into
Substituting into
We don't need to check m=-6 for that was
the value of m in part (a), and we knew
that in that case the roots were not only
reciprocals but they were also equal.
Checking m=2
,
,
So there are two answers to (c),
and .
Edwin