Try x, x+1, x+2 → x² + (x+1)² + (x+2)² = 3x² + 6x + 5 = 425 → 3x² + 6x = 420 → x² + 2x = 140 → x² + 2x − 140 = 0 → Discriminant = 564, not square. - Decision Point

Understanding the Context

When we square each term and sum them, we get:
x² + (x+1)² + (x+2)² = x² + (x² + 2x + 1) + (x² + 4x + 4)
Combine like terms:
= x² + x² + 2x + 1 + x² + 4x + 4
= 3x² + 6x + 5

This expression surprisingly equals 425, so we set up the equation:
3x² + 6x + 5 = 425

Subtract 425 from both sides to form a standard quadratic equation:
3x² + 6x + 5 − 425 = 0
→ 3x² + 6x − 420 = 0

Divide the entire equation by 3 to simplify:
x² + 2x − 140 = 0

Key Insights

Now, to solve this quadratic, apply the quadratic formula:
x = [−b ± √(b² − 4ac)] / (2a), where a = 1, b = 2, c = −140

Calculate the discriminant:
Δ = b² − 4ac = 2² − 4(1)(−140) = 4 + 560 = 564

Since 564 is not a perfect square (√564 ≈ 23.75), the solutions for x are irrational:
x = [−2 ± √564] / 2
→ x = −1 ± √141

This result reveals an important mathematical insight: while the sums of consecutive squares follow a precise pattern, arriving at a nicer number like 425 leads to a quadratic with no simple integer solutions—only irrational ones.

This example highlights how even simple patterns in algebra can lead to deeper analysis, testing both computational skill and conceptual understanding.

Final Thoughts

Conclusion:
x² + (x+1)² + (x+2)² = 425 simplifies beautifully to 3x² + 6x + 5 = 425, yielding a quadratic equation with an irrational discriminant. This reminds us that not every numerical puzzle has elegant whole-number answers—sometimes the journey reveals as much as the result.

Keywords: x² + (x+1)² + (x+2)² = 425, quadratic equation solution, discriminant 564, irrational roots, algebra pattern, math problem solving