Factoring a quadratic polynomial

Proved

WorkbookSource.problem_31836

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

For given problem, you have $ac=8$ and $bd=-5$ , Thus, $abcd=-40$ . Now, it's easy to regroup it as $ad=-20$ and $bc=2$ to make $ad+bc=-18$ . Now, $8x^2-18x-5=8x^2-20x+2x-5=(8x^2-20x)+(2x-5)=4x(2x-5)+(2x-5)=(2x-5)(4x+1)$ .

Source: InternLM Lean-Workbook, record lean_workbook_31836 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_31836  (a b c d x : ℝ)
  (h₀ : a * c = 8)
  (h₁ : b * d = -5)
  (h₂ : (a * d + b * c) = -18)
  (h₃ : x = (a + b) * (c + d)) :
  8 * x^2 - 18 * x - 5 = (2 * x - 5) * (4 * x + 1)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_31836; Apache-2.0

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