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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 1, 2024
 #1
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In order to find all of ordered pairs that work, we just have to write an equation. 

 

If the two graphs intersect, then we have \(x^3 + x^2 - 3x + 5 = x^3+2x^2\)

 

Combing all like terms and moving all terms to one side, we get \(x^2+3x-5 = 0\)

 

Using the quadratic equation, we get 

\(x=\frac{\sqrt{29}-3}{2} \approx 1.193\\ x=\frac{-\sqrt{29}-3}{2} \approx -4.193\)

 

Plugging these two values back into our equations from earlier, we get

\(y \approx 4.541 \\ y\approx -38.541\)

 

So our two orders pairs approximately are

\((1.193, 4.541), (-4.193, -38.541)\)

 

Thanks! :)

 Jun 1, 2024

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