Simplify : (6a^2-a+1)-(4-10a-3a^2)

1.	Simplify : (6a^2-a+1)-(4-10a-3a^2)
Answer» (6a²- a + 1) - (4 -10a - 3a²) = 6a²- a +1 - 4 +10a +3a² = 9a² + 9a - 3 = 9 (a² + a - \(\frac{1}{3}\)) = 9 (a² + a + \(\frac{1}{4}\) - \(\frac{1}{4}\) - \(\frac{1}{3}\)) = 9 ((a + \(\frac{1}{2}\))² - (\((\sqrt{\frac{7}{12}})^2\)) = 9 (a + \(\frac{1}{2}\) - \(\frac{\sqrt{21}}{6}\)) (a + \(\frac{1}{2}\) + \(\frac{\sqrt{21}}{6}\)) (∵ a² - b² = (a+b)(a-b) & \(\sqrt{\frac{7}{12}}\) = \(\frac{1}{2}\)\(\sqrt{\frac{7}{3}}\) = \(\frac{\sqrt 7 \times \sqrt 3}{2\sqrt 3\times \sqrt 3}\) = \(\frac{\sqrt{21}}{6}\))

Answer»

(6a²- a + 1) - (4 -10a - 3a²)

= 6a²- a +1 - 4 +10a +3a²

= 9a² + 9a - 3

= 9 (a² + a - \(\frac{1}{3}\))

= 9 (a² + a + \(\frac{1}{4}\) - \(\frac{1}{4}\) - \(\frac{1}{3}\))

= 9 ((a + \(\frac{1}{2}\))² - (\((\sqrt{\frac{7}{12}})^2\))

= 9 (a + \(\frac{1}{2}\) - \(\frac{\sqrt{21}}{6}\)) (a + \(\frac{1}{2}\) + \(\frac{\sqrt{21}}{6}\))

(∵ a² - b² = (a+b)(a-b) & \(\sqrt{\frac{7}{12}}\) = \(\frac{1}{2}\)\(\sqrt{\frac{7}{3}}\) = \(\frac{\sqrt 7 \times \sqrt 3}{2\sqrt 3\times \sqrt 3}\) = \(\frac{\sqrt{21}}{6}\))

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