SOLUTION: The number of values of a for which [(a^2)-3a+4](x^2) + [(a^2)-5a+6]x + [(a^2)-4]=0 is an identity in x is i) 0 ii) 2 iii) 1 iv) 3

Question 1074872: The number of values of a for which [(a^2)-3a+4](x^2) + [(a^2)-5a+6]x + [(a^2)-4]=0
is an identity in x is
i) 0
ii) 2
iii) 1
iv) 3
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
If that is an identities, it should be true for all values of .

For , we need to have
--> .
There are values of a that comply with that, namely and ,
but would those values of work with any and all values of ?

If for we need to have
, which simplifies to
and further simplifies to
.
That is not true for either or ,
so there are values of that make that expression an identity.
We could not find even one that would work for and ,
so none would work for every .

ANOTHER WAY:
The only way that would be an identity in x
is if ,
so the only way that
could be an identity in x for some value of a is if
has some solution.
We could have tried to find the solutions to
(no solution),
(a=3 ans a=2) , and
(a=2 and a=-2),
and then concluded that no solution worked for all 3 equations.
That may be inefficient, but it may be what was intended,
to make practice solving quadratic equations.
(Of course, if you start by finding that has no real solution,
you do not need to work on solving teh other equations
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