Answer :
To solve the given equation [tex]\(4x^4 + 5y^4 + 45x^2y^2 = 0\)[/tex] for integers [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we need to find combinations of integers that satisfy this equation.
Let's analyze the structure of this equation. The equation can be rewritten as:
[tex]\[
4x^4 + 45x^2y^2 + 5y^4 = 0
\][/tex]
Notice that each term involves squares and fourth powers, suggesting that we are dealing with a form that represents a sum of squares. For the sum of these terms to be zero, each of these terms essentially needs to cancel one another out.
### Step 1: Consider the possibility of [tex]\(x\)[/tex] or [tex]\(y\)[/tex] being zero.
1. If [tex]\(x = 0\)[/tex]:
- The equation becomes [tex]\(5y^4 = 0\)[/tex].
- Solving this, we find [tex]\(y = 0\)[/tex].
2. If [tex]\(y = 0\)[/tex]:
- The equation becomes [tex]\(4x^4 = 0\)[/tex].
- Solving this, we find [tex]\(x = 0\)[/tex].
From these results, one solution is [tex]\((x, y) = (0, 0)\)[/tex].
### Step 2: Consider non-trivial cases where neither [tex]\(x\)[/tex] nor [tex]\(y\)[/tex] is zero.
Since both variables are non-zero, let's factor the equation assuming non-zero values:
The equation remains intricate, and without complex or non-integer numbers, finding solutions is challenging. The symmetry and powers involved suggest that any proportionality in terms of non-zero [tex]\(x\)[/tex] and [tex]\(y\)[/tex] cannot satisfy the equation with real integer solutions since they introduce complex numbers when broken down.
### Conclusion:
After analyzing the equation and considering possible integer values, the only integer solution to the equation remains [tex]\((x, y) = (0, 0)\)[/tex]. For all other cases, ensuring that the equation evaluates to zero would mean involving non-real numbers, which are not integers. Therefore, the only feasible integer solution is:
[tex]\[
(x, y) = (0, 0)
\][/tex]
Let's analyze the structure of this equation. The equation can be rewritten as:
[tex]\[
4x^4 + 45x^2y^2 + 5y^4 = 0
\][/tex]
Notice that each term involves squares and fourth powers, suggesting that we are dealing with a form that represents a sum of squares. For the sum of these terms to be zero, each of these terms essentially needs to cancel one another out.
### Step 1: Consider the possibility of [tex]\(x\)[/tex] or [tex]\(y\)[/tex] being zero.
1. If [tex]\(x = 0\)[/tex]:
- The equation becomes [tex]\(5y^4 = 0\)[/tex].
- Solving this, we find [tex]\(y = 0\)[/tex].
2. If [tex]\(y = 0\)[/tex]:
- The equation becomes [tex]\(4x^4 = 0\)[/tex].
- Solving this, we find [tex]\(x = 0\)[/tex].
From these results, one solution is [tex]\((x, y) = (0, 0)\)[/tex].
### Step 2: Consider non-trivial cases where neither [tex]\(x\)[/tex] nor [tex]\(y\)[/tex] is zero.
Since both variables are non-zero, let's factor the equation assuming non-zero values:
The equation remains intricate, and without complex or non-integer numbers, finding solutions is challenging. The symmetry and powers involved suggest that any proportionality in terms of non-zero [tex]\(x\)[/tex] and [tex]\(y\)[/tex] cannot satisfy the equation with real integer solutions since they introduce complex numbers when broken down.
### Conclusion:
After analyzing the equation and considering possible integer values, the only integer solution to the equation remains [tex]\((x, y) = (0, 0)\)[/tex]. For all other cases, ensuring that the equation evaluates to zero would mean involving non-real numbers, which are not integers. Therefore, the only feasible integer solution is:
[tex]\[
(x, y) = (0, 0)
\][/tex]
