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The solutions to
2x^2 - 10x + 13 = -5x^2 - 14x - 23
are $a+bi$ and $a-bi,$ where $a$ and $b$ are positive. What is $a\cdot b?$

 Jan 8, 2024
 #1
avatar+128772 
+1

Simplify  as

 

7x^2 + 4x  + 36  =  0      divide through by 7

 

x^2  + (4/7)x + 36/7  = 0

 

x^2  + (4/7)x  =   -36/7       coplete the square on x

 

x^2 + (4/7)x  + 4/49  = -36/7 + 4/49

 

(x - 2/7)^2  =  -248/ 49          take both roots

 

x - 2/7 = [2sqrt (62) / 7] i

 

x =  2/7 + [2sqrt (62) / 7 ]  i            and   x =  2/7 - [ (2sqrt (62) / 7 ]   i

 

ab =  (2/7)(2/7)  =   4 / 49

 

cool cool cool

 Jan 8, 2024

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