Solving the Equation 16^x - 12^x - 2*9^x = 0
In this article, we will explore the solution to the equation 16^x - 12^x - 2*9^x = 0.
Step 1: Simplify the Equation
Let's start by simplifying the equation by rewriting the exponential terms using their prime factorizations:
16^x = (2^4)^x = 2^(4x) 12^x = (2^2 × 3)^x = 2^(2x) × 3^x 9^x = (3^2)^x = 3^(2x)
Now, substitute these expressions into the original equation:
2^(4x) - 2^(2x) × 3^x - 2 × 3^(2x) = 0
Step 2: Factor Out the Common Terms
Notice that 2^x is a common factor among all terms. Let's factor it out:
2^x × (2^(3x) - 2^x × 3^x - 2 × 3^(2x)) = 0
Step 3: Solve for x
Since 2^x cannot be zero (as 2 raised to any power is always positive), we can set the expression inside the parentheses equal to zero:
2^(3x) - 2^x × 3^x - 2 × 3^(2x) = 0
This is a quadratic equation in terms of 3^x. Let's make a substitution to make it easier to solve:
Let u = 3^x
Then, the equation becomes:
2^(3x) - u × 2^x - 2u^2 = 0
Substitute u = 3^x back into the equation:
2^(3x) - 3^x × 2^x - 2 × (3^x)^2 = 0
Now, we can solve for x:
x = 0 or x = 1
Conclusion
The solutions to the equation 16^x - 12^x - 2*9^x = 0 are x = 0 and x = 1.
