Find all solutions n to the equation$$6^{2n+1}+6^2=6^n+6^{n+3}.$$

3 answers
Question:

Find all solutions n to the equation
$$6^{2n+1}+6^2=6^n+6^{n+3}.$$

Answers

9514 1404 393

  n = -1 or 2

Step-by-step explanation:

The equation can be rewritten using x = 6^n. Then we can solve the quadratic.

  [tex]6^{2n+1}+6^2=6^n+6^{n+3}\\\\6\cdot(6^n)^2+6^2=6^n+6^3\cdot6^n\\\\6x^2+6^2=x(1+6^3)\\\\6x^2-6^3x-x+6^2=0\qquad\text{subtract $x(1+6^3)$}\\\\6x(x-6^2)-1(x-6^2)=0\\\\(6x-1)(x-6^2)=0\\\\x=6^n=6^{-1}\quad\text{or}\quad x=6^n=6^2\\\\\boxed{n=-1\text{ or }2}[/tex]

_____

A graphing calculator can find the zeros of the exponential function ...

  y = 6^(2n+1) +6^2 -6^n -6^(n+3) . . . . . right side subtracted


[tex]Find all solutions n to the equation $$6^{2n+1}+6^2=6^n+6^{n+3}.$$[/tex]

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