for what value of m, x4+4x2+m+8/x2+4/x4 is a complete square where x is not equal to 0
To find the value of m for which the expression is a complete square, we'll first express the given expression as a perfect square. Then we'll equate it to the square of a binomial. Here's the step-by-step process:
Step 1: Simplify the expression
Start by simplifying the expression:
x^4 + 4x^2 + (m + 8) / (x^2) + (4 / x^4)
Step 2: Combine the fractions
To combine the fractions, find the common denominator, which is x^2 * x^4 = x^6:
x^4 + 4x^2 + (m*x^6 + 8*x^4) / (x^6) + (4 / x^4)
Now, simplify the resulting expression:
x^4 + 4x^2 + (m*x^6 + 8*x^4 + 4) / (x^6)
Step 3: Express as a perfect square
To express the above expression as a perfect square, we'll try to write it in the form of (a + b)^2. Since the expression contains quadratic terms (x^4 and x^2), we can take them as the square of some variables:
Let's assume a = x^2 and b = y.
So, x^4 + 4x^2 can be written as (a^2 + 4a).
Similarly, we can write x^6 + 8x^4 + 4 as (a^3 + 8a^2 + 4).
Now, our expression becomes:
(a^2 + 4a) + (m * (a^3 + 8a^2 + 4)) / (x^6)
Step 4: Equate to a perfect square
Equating this expression to the square of a binomial:
(a + b)^2 = (a^2 + 2ab + b^2)
So, we'll set our expression equal to (a + b)^2 and expand it:
(a^2 + 4a) + (m * (a^3 + 8a^2 + 4)) / (x^6) = (a^2 + 2ab + b^2)
Cancel out the x^6 from the denominators:
(a^2 + 4a) * (x^6) + m * (a^3 + 8a^2 + 4) = (a^2 + 2ab + b^2) * (x^6)
Step 5: Compare coefficients
Now, let's compare the coefficients of the terms on both sides of the equation:
Coefficient of (x^6) on the left side: a^2 + 4a
Coefficient of (x^6) on the right side: a^2
Since the coefficients on both sides must be equal, we have:
a^2 + 4a = a^2
Simplifying, we get:
4a = 0
Since a = x^2, this implies x^2 = 0, which contradicts our given condition that x is not equal to 0.
Therefore, there is no value of m that makes the given expression a complete square.