Solve the equation $\left(2^{2 x+1}\right)\left(3^{2 x+2}\right)+2^x\left(3^{x+2}\right)-2=0$
Answer (2)
Rewrite the equation in terms of 2 x and 3 x .
Substitute u = 2 x and v = 3 x , then let w = uv to get a quadratic equation in w .
Solve the quadratic equation for w using the quadratic formula.
Substitute back to find x and verify the solution.
The solution to the equation is − 1 .
Explanation
Problem Analysis We are given the equation ( 2 2 x + 1 ) ( 3 2 x + 2 ) + 2 x ( 3 x + 2 ) − 2 = 0 and we want to solve for x .
Rewriting the Equation First, let's rewrite the equation in terms of 2 x and 3 x . We have 2 2 x + 1 = 2 ⋅ 2 2 x = 2 ⋅ ( 2 x ) 2 and 3 2 x + 2 = 3 2 ⋅ 3 2 x = 9 ⋅ ( 3 x ) 2 . Also, 3 x + 2 = 3 2 ⋅ 3 x = 9 ⋅ 3 x . Substituting these into the equation, we get ( 2 ⋅ ( 2 x ) 2 ) ( 9 ⋅ ( 3 x ) 2 ) + 2 x ( 9 ⋅ 3 x ) − 2 = 0 which simplifies to 18 ( 2 x ) 2 ( 3 x ) 2 + 9 ( 2 x ) ( 3 x ) − 2 = 0
Substitution Now, let u = 2 x and v = 3 x . The equation becomes 18 u 2 v 2 + 9 uv − 2 = 0 Let w = uv . Then the equation is 18 w 2 + 9 w − 2 = 0
Solving for w We can solve this quadratic equation for w using the quadratic formula: w = 2 a − b ± b 2 − 4 a c where a = 18 , b = 9 , and c = − 2 . Thus, w = 2 ( 18 ) − 9 ± 9 2 − 4 ( 18 ) ( − 2 ) = 36 − 9 ± 81 + 144 = 36 − 9 ± 225 = 36 − 9 ± 15 So, we have two possible values for w : w 1 = 36 − 9 + 15 = 36 6 = 6 1 w 2 = 36 − 9 − 15 = 36 − 24 = − 3 2
Solving for x Since w = uv = 2 x 3 x = 6 x , we have two equations to solve: 6 x = 6 1 6 x = − 3 2 The second equation has no real solution because 6 x is always positive. For the first equation, we have 6 x = 6 1 = 6 − 1 Therefore, x = − 1 .
Verification We can check our solution by substituting x = − 1 into the original equation: ( 2 2 ( − 1 ) + 1 ) ( 3 2 ( − 1 ) + 2 ) + 2 − 1 ( 3 − 1 + 2 ) − 2 = ( 2 − 1 ) ( 3 0 ) + ( 2 − 1 ) ( 3 1 ) − 2 = 2 1 ( 1 ) + 2 1 ( 3 ) − 2 = 2 1 + 2 3 − 2 = 2 4 − 2 = 2 − 2 = 0 Thus, our solution is correct.
Examples
Exponential equations like this one are used in various fields, such as finance to model compound interest, in biology to model population growth, and in physics to model radioactive decay. Understanding how to solve these equations allows us to predict future values based on current trends, make informed decisions about investments, and understand the behavior of natural phenomena.
The equation ( 2 2 x + 1 ) ( 3 2 x + 2 ) + 2 x ( 3 x + 2 ) − 2 = 0 can be simplified and rewritten using substitutions to find that the solution is x = − 1 . This was verified by substituting back into the original equation. Hence, the final answer is − 1 .
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