Solving Mathematical Equations: A Comprehensive Guide

Mathematics often involves solving intricate equations to find the unknown variables. This article will guide you through the step-by-step process of solving a given math problem by using both algebraic and logarithmic techniques. Understanding these methods is crucial for students, professionals, and anyone interested in mathematical problem-solving.

Solving a Quadratic Equation

Consider the equation: 2^x - 1 - 3 2^{2x} - 5. This quadratic equation can be manipulated and solved to find the value of x.

Step 1: Simplify the Equation

First, move all terms to one side to make both sides equal: 2^x - 1 - 3 2^{2x} - 5 Simplify by combining like terms: 2^x - 1 - 3 5 2^{2x} 2^x 2 2^{2x} 2^x - 2^{2x} -2

Step 2: Express as a Quadratic Equation

Rearrange the equation to match the standard form of a quadratic equation:2^{2x} - 2^{x} - 2 0 Let y 2^x, then we have:y^2 - y - 2 0

Step 3: Solve the Quadratic Equation

Solve for y using the quadratic formula:y frac{-b pm sqrt{b^2 - 4ac}}{2a} Where a 1, b -1, and c -2. Plugging in these values, we get:y frac{1 pm sqrt{1 8}}{2} frac{1 pm 3}{2}

Step 4: Determine the Value of y

Now, we have two possible values for y: 2 and -1. Since (y 2^x) and (2^x) is always positive, we discard y -1.

Step 5: Solve for x

Since (y 2^x), we have:2^x 2 Taking the logarithm base 2 of both sides:x log_2(2) 1

Alternative Methods and Complex Solutions

For a more complex solution, consider the quadratic equation 2^{2x} - 2 cdot 2^x - 8 0. Using a more advanced method, we can solve this equation by substituting (y 2^x) and solving:y^2 - 2y - 8 0

Step 1: Solve the Quadratic Equation

Using the quadratic formula:y frac{2 pm sqrt{4 32}}{2} frac{2 pm 2sqrt{9}}{2} 1 pm 3 We get:y 4, -2 Since (2^x) is always positive, we discard y -2.

Step 2: Solve for x

Since (y 2^x), we have:2^x 4 Taking the logarithm base 2 of both sides:x log_2(4) 2

Conclusion

Both methods lead us to the same solution: x 2. Understanding the steps and techniques involved in solving these equations is essential for handling more complex problems in mathematics and related fields.