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# Geometry

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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.)

Jun 19, 2024

#1
+42
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The points where the graphs intersect are points that lie on both graphs. Therefore, we can set $$x^3+x^2-3x+5=x^3+2x^2$$. Combining like terms, we get $$x^2+3x-5$$. By the quadratic formula, the roots of this equation are $$\frac{-3\pm\sqrt29}{2}$$. We have to plug these values back into one of the two equations to figure out the y coordinate. I'll let you figure that out (sry I don't have scratch paper rn).

Feel free to tell me if I did anything wrong! :D

Jun 20, 2024
#2
+129725
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Thx, Tottenham10  !!!

The solutions (per WolframAlpha) are

(  [ -3 + sqrt (29)]  / 2 , 4sqrt (29) - 17 )    and  ( [ -3 -sqrt (29)]/2 , -4sqrt (29) - 17 )

Jun 20, 2024