for what value of m, x4+4x2+m+8/x2+4/x4 is a complete square where x is not equal to 0
x^4 + 4x^2 + m + 8/x^2 + 4/x^4
well, we will have
(x^2 + k + 2/x^2)^2
x^4 + kx^2 + 2 + + kx^2 + k^2 + 2k/x^2 + 2 + 2k/x^2 + 4/x^4
x^4 + 2kx^2 + 4 + k^2 + 4k/x^2 + 4/x^4
equate like powers of x to find that we must have k=2
so m = 4+k^2 = 8
To determine the value of m for which the given expression is a complete square, we need to rewrite the expression in a perfect square form.
Let's start by reorganizing the expression:
x^4 + 4x^2 + m + (8/x^2) + (4/x^4)
Now, we can look for a perfect square expression. A perfect square expression usually represents the square of a binomial. Therefore, we can rewrite the expression as follows:
(x^2)^2 + 2(x^2)(2) + 4 + (8/x^2) + (4/x^4)
It's crucial to note that x cannot be equal to 0.
Let's focus on the terms involving x^2:
(x^2)^2 + 2(x^2)(2) + 4
This part can be written as the square of a binomial:
(x^2 + 2)^2
Now, let's look at the remaining terms:
(8/x^2) + (4/x^4)
To make this part a square of a binomial, we need to find a common denominator for the terms (8/x^2) and (4/x^4). The common denominator is x^4, so we can rewrite it as:
[(8x^2) + (4)] / x^4
At this point, we can see that (8x^2 + 4) is a perfect square expression:
(8x^2 + 4) = [(2x)^2 + 2(2x)(2) + 2^2] = (2x + 2)^2
Replacing this back into the original expression:
(x^2 + 2)^2 + [(2x + 2)^2 / x^4]
The given expression is a complete square if and only if the term [(2x + 2)^2 / x^4] equals zero.
To find the value of x, we set the numerator of this term, (2x + 2), to zero:
2x + 2 = 0
Solving for x, we subtract 2 from both sides:
2x = -2
x = -1
Since x cannot be equal to zero, the only possible value for x is -1.
Finally, we substitute x = -1 back into the expression and solve for m:
(-1^2 + 2)^2 + [(2(-1) + 2)^2 / (-1)^4] = 0
Simplifying further, we can solve for m:
(1 + 2)^2 + [(-2 + 2)^2 / 1] = 0
(3)^2 + (0)^2 = 0
9 + 0 = 0
Since this equation is not true, there is no value of m for which the given expression is a complete square when x is not equal to zero.