Solving the Equation 3×9^x-1/2-7×6^x+3×4^x+1=0
In this article, we will explore the solution to the equation 3×9^x-1/2-7×6^x+3×4^x+1=0. This equation involves exponential functions with different bases, which can make it challenging to solve. However, with the right approach, we can find the solution to this equation.
Step 1: Simplify the Equation
The first step is to simplify the equation by combining like terms. We can start by rewriting the equation as:
3×9^x - 0.5 - 7×6^x + 3×4^x + 1 = 0
Next, we can combine the constant terms:
3×9^x - 7×6^x + 3×4^x + 0.5 = 0
Step 2: Use the Properties of Exponents
To solve the equation, we need to use the properties of exponents. Specifically, we can use the rule of indices, which states that:
a^x × a^y = a^(x+y)
Using this rule, we can rewrite the equation as:
3×(3^2)^x - 7×2^x × 3^x + 3×2^2x + 0.5 = 0
Simplifying further, we get:
3^{2x+1} - 21×2^x × 3^x + 3×2^2x + 0.5 = 0
Step 3: Use the Substitution Method
At this point, it's difficult to solve the equation directly. Instead, we can use the substitution method to make the equation more manageable. Let's substitute:
u = 2^x v = 3^x
This substitution gives us:
3^{2v+1} - 21uv + 3u^2 + 0.5 = 0
Step 4: Solve the Quadratic Equation
Now, we can rewrite the equation as a quadratic equation in terms of u:
3v^2 - 21uv + 3u^2 + 0.5 = 0
Using the quadratic formula, we can solve for u:
u = (-b ± √(b^2 - 4ac)) / 2a
where a = 3, b = -21, and c = 3v^2 + 0.5.
Step 5: Back-Substitution
Once we have the value of u, we can back-substitute to find the value of x. Since u = 2^x, we can take the logarithm base 2 of both sides to get:
x = log2(u)
Similarly, we can use the value of v to find x:
x = log3(v)
Therefore, we have two expressions for x in terms of u and v.
Conclusion
Solving the equation 3×9^x-1/2-7×6^x+3×4^x+1=0 involves using the properties of exponents, substitution, and the quadratic formula. By following these steps, we can find the solution to this complex equation. However, the solution involves complex expressions involving logarithms, making it difficult to express x in a simple closed form.
