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1. The solutions of $x(x-3)=1$ 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 $abc$.

 

2. The quadratic $4x^2+2x-1$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$?

 Jan 21, 2018
 #1
avatar+98107 
+1

 

x ( x - 3)  =  1     write as

 

x^2  - 3x   -  1   =   0             complete the square on x

 

x^2  -  3x  + 9/4    -  1  -  9/4   =   0

 

(x - 3/2)^2  -  13/4  = 0

 

(x - 3/2)^2   =   13/4       take both roots

 

x -  3/2   =  ±√(13/4)

 

x -  3/2  =  ±√13 / 2

 

x  =   [ 3 ±√13 ] / 2

 

abc  =  3 * 13 *  2  =  78

 

 

cool cool cool

 Jan 21, 2018
 #2
avatar+98107 
+1

4x^2 +  2x  - 1        complete the square on x

 

4 (x ^2  +  (1/2)x   -   1/4 )

 

4( x^2  + (1/2)x + 1/16  -  1/4  -  1/16 )

 

4 (   (x + 1/4)^2   -  5/16 )

 

4(x + 1/4)^2  -  5/4

 

a  = 4      b  =  1/4       c  =  -5/4

 

a  +  b   +   c   =      4 +  1/4  - 5/4     =   3

 

 

cool cool cool

 Jan 21, 2018

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