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Find the number of real solutions to 1 + x + x^2 + x^3 = x^4 + x^5.

 Nov 11, 2019
 #1
avatar+19773 
+3

Desmos graphing show 3

 

 Nov 11, 2019
 #2
avatar+106027 
+4

1 + x + x^2 + x^3 = x^4 + x^5.


\(1 + x + x^2 + x^3 - x^4 - x^5.=0\\ (1+x)+x^2(1+x)-x^4(1+x)=0\\ (1+x)(1+x^2-x^4)=0\\ \text{one solution is x=-1}\\ Consider\;\; \\ 1+x^2-x^4=0\\ x^4-x^2-1=0\\ x^2=\frac{1\pm \sqrt{1+4}}{2}\\ x^2 \text{ can't be negative so}\\ x^2=\frac{1+ \sqrt{5}}{2}\\ x=\pm \sqrt{\frac{1+ \sqrt{5}}{2}}\;\;or\;\;-1 \)

 

So I get 3 solutions same as EP.

But I have not checked if my irrational ones are the same as his, I assum that they are.

 Nov 11, 2019
 #3
avatar+2499 
0

NOTICE HOW the value of x2 is the golden ratio!!!!

CalculatorUser  Nov 12, 2019

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