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The equation \(x^2+18x=27\) has two solutions. The positive solution has the form \(\sqrt{a}-b\) for positive natural numbers a and b. What is \(a+b\)?

 Jul 15, 2019
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x^2  +  18x  = 27       complete the square on x

 

x^2 + 18x  + 81  =  27 + 81

 

(x + 9)^2  =  108       take both roots

 

x +9  =   √108

 

x + 9   =  √[ 4 * 9 * 3]

 

x + 9  =  √4 *√9 * √3

 

x + 9   =  2 * 3 * √3

 

x + 9  =  6√3        subtract 9 from both sides

 

x  = 6√3  - 9

 

This doesn't have the specified form, Logic....but taking  "a"  as 3 and "b" as - 9  

 

a + b   =    3 + (-9)   =   -6

 

cool cool cool

 Jul 15, 2019

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