Algebra Tiles Explorations of al-Khwārizmī ’s Equation Types: Roots and Numbers are Equal to Squares

al-Khwārizmī’s sixth and final type of equation can again be modeled using either subtraction or "\(-x\)" tiles; we show only the former alternative. Because the Type 6 equation that was worked out by al-Khwārizmī (“three roots and four of simple numbers are equal to a square”) involves fractions and mixed numbers—which cannot be modeled using algebra tiles—we present a slightly-modified example in its place, following the procedure given in [Rosen 1831, 12]. We also show only the procedure for producing the positive root, as the modification for producing the negative root is analogous to how we proceeded in Example 7.

Example 10: Roots and Numbers are Equal to Squares (Type 6)