How do you factor $90 a^{2} b^{2} + 45 a b + 10$ $90a^2 b^2 +45ab+10$ ?

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How do you factor $90 a^{2} b^{2} + 45 a b + 10$ $90a^2 b^2 +45ab+10$ ?

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Answer 1 · 2017-01-27T00:15:18

Complete the square to find:

$90 a^{2} b^{2} + 45 a b + 10 = \frac{5}{8} (12 a b + 3 - \sqrt{7} i) (12 a b + 3 + \sqrt{7} i)$ $90a^2b^2+45ab+10 = 5/8(12ab+3-sqrt(7)i)(12ab+3+sqrt(7)i)$

Explanation:

$\frac{8}{5} (90 a^{2} b^{2} + 45 a b + 10) = 144 a^{2} b^{2} + 72 a b + 16$ $8/5(90a^2b^2+45ab+10) = 144a^2b^2+72ab+16$

$\frac{8}{5} (90 a^{2} b^{2} + 45 a b + 10) = {(12 a b)}^{2} + 2 (12 a b) (3) + 3^{2} + 7$ $color(white)(8/5(90a^2b^2+45ab+10)) = (12ab)^2+2(12ab)(3)+3^2+7$

$\frac{8}{5} (90 a^{2} b^{2} + 45 a b + 10) = {(12 a b + 3)}^{2} - {(\sqrt{7} i)}^{2}$ $color(white)(8/5(90a^2b^2+45ab+10)) = (12ab+3)^2-(sqrt(7)i)^2$

$\frac{8}{5} (90 a^{2} b^{2} + 45 a b + 10) = (12 a b + 3 - \sqrt{7} i) (12 a b + 3 + \sqrt{7} i)$ $color(white)(8/5(90a^2b^2+45ab+10)) = (12ab+3-sqrt(7)i)(12ab+3+sqrt(7)i)$

So:

$90 a^{2} b^{2} + 45 a b + 10 = \frac{5}{8} (12 a b + 3 - \sqrt{7} i) (12 a b + 3 + \sqrt{7} i)$ $90a^2b^2+45ab+10 = 5/8(12ab+3-sqrt(7)i)(12ab+3+sqrt(7)i)$

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How do you factor $90 a^{2} b^{2} + 45 a b + 10$ $90a^2 b^2 +45ab+10$ ?

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