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There are two distinct solutions $x$ to the equation $18+5x^2=20x$. If each solution is rounded to the nearest integer, and then these two integers are multiplied together, what is the result?

Guest Jan 21, 2018
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18 + 5x^2  =  20x     rewrite as

 

5x^2 -  20x +  18  =  0

 

Complete the square on x

 

5 ( x^2   -  4x  +  18/5)  = 0

 

5 ( x^2  -  4x   +  4   +  18/5  -  4)  = 0

 

5 (  (x - 2)^2   +  18/5  - 20/5 )   =   0

 

5( (x - 2)^2  - 2/5)  = 0

 

(x - 2)^2  -  2/5  =   0

 

(x - 2) ^2   =  2/5           take both roots

 

x -2  = ±√(2/5)

 

x  =    2 ±√(2/5)

 

This evaluates to     x  ≈  1.3675    x  ≈ 2.6325

 

The desired rounding produces     1   and 3     and their product is   3

 

 

cool cool cool

CPhill  Jan 21, 2018

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