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avatar+1791 

What are the coordinates of the points where the graphs  of f(x)=x^3 + x^2 - 3x + 5 and g(x) = x^3 + 2x^2 intersect?  

 

Give your answer as a list of points separated by commas, with the points ordered such that their -coordinates are in increasing order. (So "(1,-3), (2,3), (5,-7)" - without the quotes - is a valid answer format.)

 Jan 19, 2024
 #1
avatar+128796 
+1

x^3 + x^2 -3x + 5 = x^3 + 2x^2

 

x^2 + 3x -  5   = 0

 

x^2 + 3x  =  5            complete the square on x

 

x^2 + 3x + 9/4  = 5 + 9/4

 

(x + 3/2)^2  = 29/4           take  both roots

 

 

x + 3/2 =  sqrt (29/4)                       x + (3/2)  = -sqrt (29/4)

x = [sqrt (29 - 3 ] / 2                        x  =   [ -sqrt (29) - 3 ] / 2

y = [-sqrt (29) - 3 ] / 2                       y =    [sqrt (29) -3 ]  / 2

 

(x,y)  = ( [sqrt (29 -3 ]  /2 , [-sqrt (29)-3 ] /2 ,  [-sqrt (29) -3 ] / 2  , [ sqrt (29) - 3 ] / 2 )

 

 

cool cool cool

 Jan 19, 2024

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