We try all integer \( x \) from \( -4 \) to \( 4 \), compute \( 9x^2 \), then \( 16y^2 = 144 - 9x^2 \), so \( y^2 = \frac{144 - 9x^2}{16} \), must be a perfect square and integer.

["Title: Solve the Integer Equation: Testing All ( x ) from (-4) to (4) for ( y^2 = \frac{144 - 9x^2}{16} ) to Be a Perfect Square", "---", "### Introduction\nMathematics often reveals elegant connections between equations, and one fascinating exercise is solving a systematic case by testing integer values across a bounded range. In this article, we explore the equation:\n[\n9x^2 \quad \ ext{computed for } x \in {-4, -3, -2, -1, 0, 1, 2, 3, 4},\n]\nand then compute ( y^2 = \frac{144 - 9x^2}{16} ), checking whether ( y^2 ) is a perfect square and an integer. This approach helps identify valid solutions under realistic constraints.", "---", "### Step-by-Step Computation\nThe key formula is:\n[\ny^2 = \frac{144 - 9x^2}{16}.\n]\nWe evaluate this expression for each integer ( x ) between (-4) and (4), ensure ( y^2 ) is nonnegative, and verify it’s a perfect square.", "---", "#### For ( x = -4 ) and ( x = 4 ):\n( x^2 = 16 ), so\n[\ny^2 = \frac{144 - 9 \cdot 16}{16} = \frac{144 - 144}{16} = 0.\n]\n( y^2 = 0 ) is a perfect square (( y = 0 )) and an integer. ✅ Valid solution.", "---", "#### For ( x = -3 ) and ( x = 3 ):\n( x^2 = 9 ), so\n[\ny^2 = \frac{144 - 81}{16} = \frac{63}{16} \approx 3.9375.\n]\nNot an integer. ❌ No valid ( y ).", "---", "#### For ( x = -2 ) and ( x = 2 ):\n( x^2 = 4 ), so\n[\ny^2 = \frac{144 - 36}{16} = \frac{108}{16} = 6.75.\n]\nNot an integer. ❌ No valid ( y ).", "---", "#### For ( x = -1 ) and ( x = 1 ):\n( x^2 = 1 ), so\n[\ny^2 = \frac{144 - 9}{16} = \frac{135}{16} = 8.4375.\n]\nNot an integer. ❌ No valid ( y ).", "---", "#### For ( x = 0 ):\n( x^2 = 0 ), so\n[\ny^2 = \frac{144 - 0}{16} = \frac{144}{16} = 9.\n]\n( y^2 = 9 ) is a perfect square (( y = 3 ) or ( y = -3 )) and an integer. ✅ Valid solution.", "---", "### Summary of Valid Solutions\nOnly when ( x = -4, 0, 4 ) does ( y^2 ) become a nonnegative perfect square:", "| ( x ) | ( x^2 ) | ( 9x^2 ) | ( 144 - 9x^2 ) | ( y^2 = \frac{144 - 9x^2}{16} ) | Integer? | Perfect Square? |\n|--------|----------|-----------|-----------------|----------------------------------|----------|----------------|\n| -4 | 16 | 144 | 0 | 0 | Yes | ( 0^2 ) |\n| 0 | 0 | 0 | 144 | 9 | Yes | ( 3^2 ) |\n| 4 | 16 | 144 | 0 | 0 | Yes | ( 0^2 ) |", "---", "### Why This Method Works\nThis brute-force check within a fixed integer range ensures no case is missed and verifies the critical condition: ( y^2 ) must be both an integer and a perfect square for divisibility and solution validity. This process strengthens number-theoretic reasoning when constraints are known.", "---", "### Real-World Applications\nSuch integer tests appear in modular arithmetic, Diophantine equations, cryptography, and algorithm validation—especially when searching for lattice points or discrete solutions. Programs dealing with polynomial constraints often implement similar integer sweeps for validation.", "---", "### Conclusion\nTesting all integer values from (-4) to (4) in the expression ( y^2 = \frac{144 - 9x^2}{16} ) reveals that only specific values of ( x )—(-4, 0, 4)—produce integer and perfect square values for ( y^2 ). This method highlights the power of systematic testing in discrete math and computational verification.", "For anyone solving equations with integer constraints, bounded enumeration remains a simple yet effective strategy. Run your own cases to deepen understanding!", "---", "Keywords: integer solutions, ( y^2 = \frac{144 - 9x^2}{16} ), perfect square test, bounded computation, Diophantine equation, valid ( x ), computational verification.", "---", "Meta Description:\nTest all ( x ) from (-4) to (4), compute ( y^2 = \frac{144 - 9x^2}{16} ), and verify if it's a perfect integer square. Discover valid solutions through systematic enumeration—ideal for discrete math and algorithm prep."]