1Cademy - Simplifying $$(6-\sqrt{5})^2$$ and $$(9-2\sqrt{10})^2$$

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Binomial Squares Pattern

Example

Simplifying $(6-\sqrt{5})^2$ and $(9-2\sqrt{10})^2$

Expand these expressions by applying the Binomial Squares Pattern for a difference: $(a - b)^2 = a^2 - 2ab + b^2$ .

For $(6 - \sqrt{5})^2$ , let $a = 6$ and $b = \sqrt{5}$ : $6^2 - 2(6)(\sqrt{5}) + (\sqrt{5})^2 = 36 - 12\sqrt{5} + 5$ Combine the numerical terms to get $41 - 12\sqrt{5}$ .

For $(9 - 2\sqrt{10})^2$ , let $a = 9$ and $b = 2\sqrt{10}$ : $9^2 - 2(9)(2\sqrt{10}) + (2\sqrt{10})^2$ Squaring the second term requires squaring both the coefficient and the radical: $2^2 \cdot (\sqrt{10})^2 = 4 \cdot 10 = 40$ . The expanded expression is: $81 - 36\sqrt{10} + 40$ Combine the numerical terms to arrive at the final result: $121 - 36\sqrt{10}$ .

Updated 2026-05-01

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OpenStax Intermediate Algebra

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