Problem 33 - Algebra and Trigonometry

Question

Solve the equation. \(x^{2} 2^{x}-2^{x}=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -1\).

Step 1: Factor Common Terms

First, observe the given equation \(x^{2} 2^{x} - 2^{x} = 0\). Notice that \(2^{x}\) is a common factor in both terms. We can factor \(2^{x}\) out of the equation: \[ 2^{x}(x^2 - 1) = 0 \]

Step 2: Set Factors to Zero

For the product of two terms to be zero, at least one of them must be zero. So, we set each factor equal to zero separately:1. \(2^{x} = 0\)2. \(x^2 - 1 = 0\)

Step 3: Solve the First Factor

Consider the first factor: \(2^{x} = 0\). Since \(2^{x}\) is never zero for any real number \(x\), this factor does not provide any solutions.

Step 4: Solve the Second Factor

Now, solve the second factor: \(x^2 - 1 = 0\).This equation can be rewritten as:\[ x^2 = 1 \]Solving for \(x\), we take the square root of both sides, yielding:\[ x = \pm 1 \]

Step 5: Combine Solutions

The solutions from the equation are those obtained from the second factor, which are \(x = 1\) and \(x = -1\). Thus, these are the solutions for the equation given.

Key Concepts

Factoring

Factoring is a powerful mathematical method used to simplify equations, especially quadratic ones. In the given equation, \(x^2 2^x - 2^x = 0\), the first step is to identify common factors. Here, \(2^x\) appears in both terms on the left-hand side. By factoring out \(2^x\), the equation becomes \(2^x(x^2 - 1) = 0\). This simplification helps in identifying the parts of the equation we need to investigate further. When you factor an expression, you rewrite it as a product of its simplest form. This process reveals its roots or solutions more clearly. Factoring relies on identifying commonalities within terms, making it easier to set up simpler equations that are often easier to solve. Checklist for factoring:

• Look for common factors shared among terms.
• Rewrite the equation as a product.
• Simplify further if possible by factoring smaller components.

By applying these steps, you can break down complex problems into manageable ones.

Equation Solving

Equation solving is a critical skill in mathematics, involving finding values that make the equation true. Once we've factored our original equation into \(2^x(x^2 - 1) = 0\), we know that for the product to be zero, at least one of the factors must be zero.Let's walk through the steps:

• Set \(2^x = 0\). This doesn't work because exponential functions \(2^x\) never equal zero for any real \(x\).
• Next up is \(x^2 - 1 = 0\), which simplifies to \(x^2 = 1\). Solving this involves taking the square root of both sides, yielding \(x = \pm 1\).

The concept of solving equations involves recognizing how to break down and manage different components. Focus on each segment individually to understand their properties and how they contribute to the solution.

Exponential Functions

Exponential functions like \(2^x\) are crucial in various mathematical contexts. In our context, it's important to remember that \(2^x\) represents continuous growth and is never zero for any real number \(x\). Exponential functions are defined mathematically as \(a^x\) where \(a\) is a positive constant. These functions do not cross the x-axis, meaning they never equate to zero. This property is consistent with the base of the exponent (which is 2 in this case) being constant.Understanding exponential functions involves:

• Recognizing their patterns of growth or decay.
• Identifying intercepts and asymptotes.
• Practicing transformations such as shifts and stretches.

These functions appear frequently not just in equations but also in modeling real-world phenomena, emphasizing their importance in both math and applied sciences.