1.3 Square root of a quadratic surd
( a + b ) 2 = a 2 + b 2 + 2 a b (a + b)^2 = a^2 + b^2 + 2ab ( a + b ) 2 = a 2 + b 2 + 2 ab ( a 2 + b 2 + 2 a b ) (a^2 + b^2 + 2ab) ( a 2 + b 2 + 2 ab ) . a 2 + b 2 a^2 + b^2 a 2 + b 2 forms the rational part and 2 a b 2ab 2 ab forms the irrational part.
Example 4
Find the square root of 9 + 4 5 9 + 4 \sqrt{5} 9 + 4 5 Solution
9 + 4 5 = 9 + 2 20 9 + 4 \sqrt{5} = 9 + 2 \sqrt{20} 9 + 4 5 = 9 + 2 20 = 4 + 5 + 2 4 × 5 = 4 + 5 + 2 \sqrt{{4} \times {5}} = 4 + 5 + 2 4 × 5 = ( 2 ) 2 + ( 5 ) 2 + 2 × 2 × 5 = (2)^2 + (\sqrt{5})^2 + 2 \times 2 \times \sqrt{5} = ( 2 ) 2 + ( 5 ) 2 + 2 × 2 × 5 = ( 2 + 5 ) 2 = (2 + \sqrt{5})^2 = ( 2 + 5 ) 2 ∴ 9 + 4 5 = ( 2 + 5 ) 2 = ± ( 2 + 5 ) \therefore \sqrt{{9 + 4 \sqrt{5}}} = \sqrt{{(2 + \sqrt {5})^2}} = \pm(2 + \sqrt{5}) ∴ 9 + 4 5 = ( 2 + 5 ) 2 = ± ( 2 + 5 ) Answer : ± ( 2 + 5 ) \pm(2 + \sqrt{5}) ± ( 2 + 5 )
Example 5
Find the square root of 15 − 4 14 15 - 4 \sqrt{14} 15 − 4 14 7 − 2 2 \sqrt{7} - 2\sqrt{2} 7 − 2 2 (2) 7 − 6 \sqrt{7} - \sqrt{6} 7 − 6 (3) 8 − 7 \sqrt{8} - \sqrt{7} 8 − 7 (4) More than one of the above Solution
Let x = 15 − 4 14 = 15 − 2 56 x = 15 - 4 \sqrt{14} = 15 - 2 \sqrt{56} x = 15 − 4 14 = 15 − 2 56 = 7 + 8 − 2 7 × 8 = 7 + 8 - 2 \sqrt{{7} \times {8}} = 7 + 8 − 2 7 × 8 = ( 7 − 8 ) 2 = (\sqrt{7} - \sqrt{8})^2 = ( 7 − 8 ) 2 x = ( 7 − 8 ) 2 = ± ( 7 − 8 ) \sqrt{x} = \sqrt{(\sqrt{7} - {8})^2} = \pm(\sqrt{7} - \sqrt{8}) x = ( 7 − 8 ) 2 = ± ( 7 − 8 ) 7 − 2 2 = 7 − 8 \sqrt{7} - 2\sqrt{2} = \sqrt{7} - \sqrt{8} 7 − 2 2 = 7 − 8 , which is satisfied 8 − 7 = − ( 7 − 8 ) \sqrt{8} - \sqrt{7} = - (\sqrt{7} - \sqrt{8}) 8 − 7 = − ( 7 − 8 ) , which is also satisfiedAnswer : (4) More than one of the above
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