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The roots of the quadratic equation $x(x-3)=1+8x-5$ may be expressed in the form $\frac{a+\sqrt{b}}{c}$ and $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are prime numbers.  Find $abc$.

 Aug 27, 2023
 #1
avatar+128732 
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x(x-3)=1 + 8x - 5

 

x^2 - 3x  = -4 + 8x

 

x^2 - 11x  =  -4       complete the square on x

 

x^2 - 11x + 121/4 =  -4 + 121/4

 

(x  -  11/2)^2  =  -16/4 + 121/4

 

(x - 11/2)^2  = 105/4      we only need  one  root  to  find a,b,c

 

x - (11/2)  =  sqrt (105) / 2

 

x =  [ 11 + sqrt (105) ] /  2

 

 

abc =   (11)(105)(2) =  2310

 

 

cool cool cool

 Aug 27, 2023

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