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avatar+1616 

What are the coordinates of the points where the graphs  of $f(x)=x^3 + x^2 - 3x + 5$ and $g(x)=x^4-7x^3+5x^2-18x+17$ intersect?  

 

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

 Dec 2, 2023
 #1
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By the quadratic formula, the roots of x2−27​x+413​ are [ \frac{7\pm\sqrt{\left(\frac{7}{2}\right)^2-4\cdot1\cdot\frac{13}{4}}}{2}=\frac{7\pm\sqrt{1}}{2}=\frac{7}{2}\pm\frac{\sqrt{1}}{2} ]

 

Therefore, the values of x at which the graphs of f and g intersect are x=2, x=27+1​​ and x=27−1​​.

 

We can plug these x values back into f(x) to get that f(2)=21, f(27+1​​)=825+1​​ and f(27−1​​)=825−1​​, so the points of intersection are [ \left(2,21\right),\left(\frac{7+\sqrt1}{2},\frac{25+\sqrt{1}}{8}\right),\left(\frac{7-\sqrt1}{2},\frac{25-\sqrt{1}}{8}\right). ]

 Dec 2, 2023

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