Try x, x+1, x+2: Deriving the Truth Behind x² + (x+1)² + (x+2)² = 425

Mathematics is full of elegant patterns and clever shortcuts, and one fascinating expression involves squaring three consecutive integers:

x, x+1, x+2

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

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.

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.

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Year 1: 6
Year 2: 22
Year 3: 38

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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