For each \( x \), check if \( 225 - 9x^2 \) is divisible by 25 and the result for \( y^2 \) is a perfect square.

["SEO Article: Checking When ( 225 - 9x^2 ) is Divisible by 25 — When Is ( y^2 ) a Perfect Square?", "---", "Understanding the Problem: For Each ( x ), Evaluate ( 225 - 9x^2 ) Modulo 25 and Analyze Perfect Square Conditions", "Mathematics often challenges us to explore relationships between quadratic expressions and divisibility. One fascinating question arises in algebra:\nFor each integer ( x ), check if ( 225 - 9x^2 ) is divisible by 25 and whether the resulting value makes ( y^2 = 225 - 9x^2 ) a perfect square.", "This article explores this problem step-by-step—understanding the conditions, applying modular arithmetic, and identifying values of ( x ) that satisfy both divisibility and perfect square requirements. It’s a great exercise for algebra students, enthusiasts, and educators alike.", "---", "### What Does It Mean for ( 225 - 9x^2 ) to Be Divisible by 25?", "We want to determine when:\n[\n225 - 9x^2 \equiv 0 \pmod{25}\n]", "This means we seek integer solutions ( x ) satisfying:\n[\n9x^2 \equiv 225 \pmod{25}\n]", "Since ( 225 \div 25 = 9 ), we simplify:\n[\n9x^2 \equiv 0 \pmod{25}\n]", "Because 9 and 25 are coprime (gcd(9,25) = 1), we can divide both sides of the congruence by 9 modulo 25:\n[\nx^2 \equiv 0 \pmod{25}\n]", "This implies:\n[\nx \equiv 0 \pmod{5}\n]", "So, for divisibility by 25, ( x ) must be a multiple of 5.", "---", "### Restricting ( x ): Symbolic Form and Perfect Square Check", "Let ( x = 5k ), where ( k ) is any integer. Then substitute into the expression:\n[\n225 - 9x^2 = 225 - 9(5k)^2 = 225 - 9 \cdot 25k^2 = 225 - 225k^2 = 225(1 - k^2)\n]", "We want this to be a perfect square:\n[\ny^2 = 225(1 - k^2)\n]", "Since ( 225 = 15^2 ), we rewrite:\n[\ny^2 = 15^2(1 - k^2) = (15\sqrt{1 - k^2})^2\n]", "For ( y^2 ) to be a perfect square, ( 1 - k^2 ) must also be a perfect square times a rational square, but since we seek integer ( y ), ( 1 - k^2 ) must be a rational square — and more specifically, since ( y^2 ) is integer, ( 1 - k^2 ) must be a perfect square of a rational number. However, ( k ) is integer, so ( 1 - k^2 ) is integer.", "Therefore, ( 1 - k^2 ) must be a non-negative perfect square.", "Let:\n[\n1 - k^2 = m^2,\quad m \in \mathbb{Z},\quad m^2 \geq 0\n]", "Then:\n[\nk^2 + m^2 = 1\n]", "The only integer solutions to this Diophantine equation are:\n- ( k = \pm1, m = 0 )\n- ( k = 0, m = \pm1 )", "But since ( m^2 = 1 - k^2 \geq 0 ), valid pairs:\n- ( k^2 = 0 \Rightarrow k = 0 \Rightarrow m^2 = 1 ) → ( y^2 = 225 \cdot 1 = 225 = 15^2 ) ✅\n- ( k^2 = 1 \Rightarrow k = \pm1 \Rightarrow m^2 = 0 \Rightarrow y^2 = 0 = 0^2 ) ✅\n- ( k^2 > 1 \Rightarrow 1 - k^2 < 0 ) → invalid", "Also check ( k = 0 ): then ( x = 5k = 0 ), gives ( y^2 = 225 ) ✅\nCheck ( k = \pm1 ): ( x = 5(\pm1) = \pm5 ), then:\n[\ny^2 = 225(1 - 1) = 0 = 0^2 ] ✅", "---", "### Valid Values of ( x ) and Resulting ( y^2 )", "| ( x ) | ( x^2 ) | ( 9x^2 ) | ( 225 - 9x^2 ) | Divisible by 25? | ( y^2 ) | Is Perfect Square? |\n|-------------|----------|------------|------------------|------------------|----------|--------------------|\n| 0 | 0 | 0 | 225 | Yes | 225 | Yes (15²) |\n| ±5 | 25 | 225 | 0 | Yes | 0 | Yes (0²) |", "For ( k = \pm2 ), ( x = \pm10 ):\n( x^2 = 100 ), ( 9x^2 = 900 ), ( 225 - 900 = -675 ) — negative, not allowed in standard perfect square interpretation (though algebraic, not a positive square)", "---", "### Conclusion: Final Answer and Takeaways", "Summary: For each integer ( x ),\n- ( 225 - 9x^2 ) is divisible by 25 if and only if ( x ) is a multiple of 5: ( x = 5k ).\n- Under this condition, ( y^2 = 225 - 9x^2 = 225(1 - k^2) )\n- This is a perfect square only when ( 1 - k^2 ) is a perfect square, which occurs only when ( k = 0 ) or ( k = \pm1 )", "Thus, the only integer values of ( x ) satisfying both conditions are:\n[\nx = 0, \pm5\n]\nfor which ( y^2 ) is a perfect square.", "---", "### Why This Matters", "This problem blends modular arithmetic, substitution, and number theory. It helps students:\n- Understand conditions under which expressions yield integer results\n- Explore divisibility beyond simple division\n- Recognize the special role of perfect squares in algebra", "Whether for homework, exams, or deepening mathematical intuition—this kind of structured analysis is invaluable.", "---", "Keywords:\n( 225 - 9x^2 ), divisibility by 25, perfect square, modular arithmetic, ( x ) multiple of 5, ( y^2 = 225 - 9x^2 ), algebra problem, integer solutions, number theory.", "Meta Description:\nFor each ( x ), check whether ( 225 - 9x^2 ) is divisible by 25 and if ( y^2 = 225 - 9x^2 ) is a perfect square. Explore conditions, integer solutions, and algebraic reasoning using modular arithmetic.", "---", "Explore more at [Your Math Learning Site] – where math meets clarity!", "---", "Internal Links:\n- Perfect Squares and Divisibility Rules\n- Modular Arithmetic for Algebra Students\n- How to Solve Quadratic Diophantine Equations", "---", "© 2024 Algebra Insights – Certified educational resource."]