Solving the Equation: 2^x + 10/4 = 9/2^x - 2
In this article, we will solve the equation 2^x + 10/4 = 9/2^x - 2. This equation involves exponential functions and fractions, making it a bit more challenging to solve. However, with the right steps, we can find the value of x.
Step 1: Simplify the Equation
First, let's simplify the equation by combining the fractions:
2^x + 2.5 = 9/2^x - 2
Step 2: Multiply Both Sides by 2^x
To eliminate the fraction on the right side of the equation, let's multiply both sides by 2^x:
2^x * 2^x + 2.5 * 2^x = 9 - 2 * 2^x
This gives us:
2^(2x) + 2.5 * 2^x = 9 - 2 * 2^x
Step 3: Rearrange the Terms
Rearrange the terms to get a quadratic equation in 2^x:
2^(2x) + 4.5 * 2^x - 9 = 0
Step 4: Factor the Equation (Optional)
If possible, we can factor the equation:
(2^x + 9)(2^x - 1) = 0
This gives us two possible solutions:
2^x + 9 = 0 --> 2^x = -9 (no real solution)
2^x - 1 = 0 --> 2^x = 1 --> x = 0
Step 5: Solve the Quadratic Equation
If factoring is not possible, we can use the quadratic formula:
2^x = (-4.5 ± √(4.5^2 - 4 * 1 * (-9))) / 2
2^x = (-4.5 ± √(20.25 + 36)) / 2
2^x = (-4.5 ± √56.25) / 2
2^x = (-4.5 ± 7.5) / 2
This gives us two possible solutions:
2^x = (-4.5 + 7.5) / 2 --> 2^x = 1.5 --> x = log2(1.5)
2^x = (-4.5 - 7.5) / 2 --> 2^x = -6 --> no real solution
Conclusion
Therefore, the solution to the equation 2^x + 10/4 = 9/2^x - 2 is x = log2(1.5).
