Solving the Equation 4^3x^2+x-8=2*8^x^2+x/3
In this article, we will solve the equation 4^3x^2+x-8=2*8^x^2+x/3. This equation involves exponential functions, polynomial terms, and fractional terms. To solve this equation, we need to use a combination of algebraic manipulations and mathematical techniques.
Step 1: Simplify the Equation
First, let's simplify the equation by evaluating the exponents and multiplying the terms:
4^3x^2 = 64x^2 (since 4^3 = 64)
28^x^2 = 2(8^x^2) = 16^x^2 (since 28 = 16)
Now the equation becomes:
64x^2 + x - 8 = 16^x^2 + x/3
Step 2: Move All Terms to One Side
Next, let's move all terms to the left side of the equation:
64x^2 + x - 8 - 16^x^2 - x/3 = 0
Step 3: Factor Out the Greatest Common Factor (GCF)
Notice that the GCF of the terms on the left side is x^2. Let's factor it out:
x^2(64 + 1 - 16^x - 1/3) = 0
Step 4: Solve for x
Now we have a quadratic equation in terms of x^2. Let's set the expression inside the parentheses equal to 0 and solve for x:
64 + 1 - 16^x - 1/3 = 0
Simplifying the equation, we get:
65 - 16^x - 1/3 = 0
Multiplying both sides by 3 to eliminate the fraction, we get:
195 - 48^x - 1 = 0
Rearranging the terms, we get:
48^x = 194
Step 5: Take the Logarithm
To solve for x, we can take the logarithm of both sides:
x^2 * log(48) = log(194)
Dividing both sides by log(48), we get:
x^2 = log(194) / log(48)
Step 6: Find the Square Root
Finally, we take the square root of both sides to find x:
x = ±√(log(194) / log(48))
Therefore, we have found the solutions to the equation 4^3x^2+x-8=2*8^x^2+x/3.
Conclusion
In this article, we solved the equation 4^3x^2+x-8=2*8^x^2+x/3 using a combination of algebraic manipulations, exponential functions, and mathematical techniques. We showed that the solution involves taking the logarithm and finding the square root.
