$(2^{2 x+1})(3^{2 x+2})+2^n(3^{x+2})-2=0

Rewrite the given equation in terms of exponential functions.
Substitute y = 3 x and z = 2 x to simplify the equation into a quadratic form.
Solve the quadratic equation for y in terms of z and n , choosing the positive root since 0"> y > 0 .
Substitute back to find the relationship between x and n : 3 x = 36 × 2 2 x − 9 × 2 n + 81 × 2 2 n + 144 × 2 2 x ​ ​ .
When n = − 1 , x = − 1 is a solution, which can be verified by substitution.
The final relationship is: 3 x = 36 × 2 2 x − 9 × 2 n + 81 × 2 2 n + 144 × 2 2 x ​ ​ ​

Explanation

Problem Analysis We are given the equation ( 2 2 x + 1 ) ( 3 2 x + 2 ) + 2 n ( 3 x + 2 ) − 2 = 0 and we want to find the relationship between x and n that satisfies this equation.

Rewriting the Equation First, let's rewrite the equation to make it easier to work with: ( 2 2 x × 2 ) ( 3 2 x × 3 2 ) + 2 n ( 3 x × 3 2 ) − 2 = 0 2 × 9 × ( 2 2 × 3 2 ) x + 2 n × 9 × 3 x − 2 = 0 18 × ( 36 ) x + 9 × 2 n × 3 x − 2 = 0

Substitution Let y = 3 x . Then we can rewrite the equation as: 18 × ( 4 x ) × ( 3 x ) 2 + 9 × 2 n × 3 x − 2 = 0 18 × ( 4 x ) × y 2 + 9 × 2 n × y − 2 = 0

Further Substitution Let z = 2 x . Then the equation becomes: 18 × ( 2 x ) 2 × y 2 + 9 × 2 n × y − 2 = 0 18 z 2 y 2 + 9 × 2 n y − 2 = 0 This is a quadratic equation in terms of y .

Solving for y Now, we solve the quadratic equation for y in terms of z and n :
y = 2 ( 18 z 2 ) − 9 × 2 n ± ( 9 × 2 n ) 2 − 4 ( 18 z 2 ) ( − 2 ) ​ ​ y = 36 z 2 − 9 × 2 n ± 81 × 2 2 n + 144 z 2 ​ ​

Choosing the Positive Root Since 0"> y = 3 x > 0 , we take the positive root: y = 36 z 2 − 9 × 2 n + 81 × 2 2 n + 144 z 2 ​ ​

Final Relationship Substitute back y = 3 x and z = 2 x to obtain the relationship between x and n :
3 x = 36 × 2 2 x − 9 × 2 n + 81 × 2 2 n + 144 × 2 2 x ​ ​

Verification From the python calculation tool, we found that when n = − 1 , x = − 1 is a solution. Let's verify this: ( 2 2 ( − 1 ) + 1 ) ( 3 2 ( − 1 ) + 2 ) + 2 − 1 ( 3 − 1 + 2 ) − 2 = ( 2 − 1 ) ( 3 0 ) + ( 1/2 ) ( 3 ) − 2 = ( 1/2 ) ( 1 ) + 3/2 − 2 = 1/2 + 3/2 − 4/2 = 0 So, x = − 1 when n = − 1 is indeed a solution.


Examples
This type of equation, involving exponential terms, can be used to model various phenomena in physics and engineering, such as radioactive decay or the growth of bacterial populations. By understanding the relationship between the variables, we can predict the behavior of these systems over time. For instance, in finance, similar equations can model the growth of investments with compound interest, where the exponent represents time and the base represents the growth factor.

Answered by GinnyAnswer | 2025-07-04

To solve the equation ( 2 2 x + 1 ) ( 3 2 x + 2 ) + 2 n ( 3 x + 2 ) − 2 = 0 , we rewrite it into a quadratic form and substitute variables. After deriving the positive root given the parameters of the equation, we find a relationship between x and n and verify specific solutions. The final equation links the variables through their exponential forms, enhancing our understanding of their interconnected behavior.
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Answered by Anonymous | 2025-07-05