Question

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

Answer 1

9514 1404 393

Answer:

n = -1 or 2

Explanation:

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

`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}`

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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