MathLabs

Problem 2

Find all natural numbers xx such that the product of their digits (in decimal notation) is equal to x2−10x−22x^2 - 10x - 22.
Step 2 of 5: The equation forces x ≤ 12
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In plain words

This is the standard move for equations mixing a fast-growing quadratic with a slow-growing, bounded quantity: turn the bound into an inequality, and the infinite search collapses to a finite, checkable list.

x2−10x−22≤x  ⟹  x2−11x−22≤0  ⟹  1≤x≤12x^2-10x-22 \le x \;\Longrightarrow\; x^2-11x-22\le 0 \;\Longrightarrow\; 1\le x\le 12
Detailed analysis

If xx satisfies the problem's equation, its digit product equals x2−10x−22x^2-10x-22, and by Step 1 this digit product is at most xx. So x2−10x−22≤xx^2-10x-22\le x, i.e. x2−11x−22≤0x^2-11x-22\le0. The quadratic t2−11t−22t^2-11t-22 has roots 11±2092\dfrac{11\pm\sqrt{209}}{2}, and 209≈14.46\sqrt{209}\approx14.46, so the inequality holds exactly for 11−2092≤x≤11+2092≈12.73\dfrac{11-\sqrt{209}}{2}\le x\le\dfrac{11+\sqrt{209}}{2}\approx12.73. Since xx is a natural number, this leaves only x∈{1,2,…,12}x\in\{1,2,\dots,12\} to check.