the value of n for which the expression [9x^4 - 12x^2 + nx^2 - 8x + 4] be a perfect sqaure? the options are:?

Question

1)12
2)16
3)18
4)24

the right option is 16.. please show ur working out:)

Answer 1

if this is a perfect square it is perfect square for any x .

say 1
9-12+n-8+4 or n-7 is a pefect square

so n has to be 16 for 16-7 is a perfect square

Answer 2

You've written -12x^2, which I think should be -12x^3.
So we have : 9x^4 - 12x^3 + nx^2 - 8x + 4.

Look at the first and last terms.
The square root of 9x^4 is 3x^2 and
the square root of 4 is 2.
We'll also probably need a term in x.
Let it be mx where m is unknown at the moment.

So your expression could equal (3x^2 + mx + 2)^2,
which is a square.

Expanding gives : 9x^4 + m^2x^2 + 4 + 6mx^3 + 12x^2 + 4mx

Adding like terms and rearranging gives :
9x^4 + 6mx^3 + (m^2+ 12)x^2 + 4mx + 4

This now looks similar to your expression of :
9x^4 - 12x^3 + nx^2 - 8x + 4

Equating powers of x gives :

For x^3 we have : 6m = -12, or, m = -2.
For x we have : 4m = -8. Again, m = -2.

For x^2 we have : m^2 + 12 = n.
Knowing now that m = -2, substitute this here to get :
(-2)^2 + 12 = n
Therefore, n = 4 + 12 = 16.

Answer 3

consider 9x^4-12x^2+nx^2-8x+4
=9x^4+(n-12)x^2-8x+4 -----------(1)
if n=12,18,24
then (1) becomes 9x^4-8x+4,9x^4+6x^2-8x+4,9x^4+12x^2-8x+4 respectively.
but this expression cannot be a perfect square according to its coefficient and powers of x

if n=16
then 9x^4+4x^2-8x+4 is the perfect square

Answer 4

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Answer 5

I don't think so. This looks like a homework question and YOU really do need to learn how to do this. I already learned it. Now it is YOUR turn.