Problem 25 - Intermediate Algebra

Question

Solve each equation. $$ 2^{x^{2}-2 x}=8 $$

Step-by-Step Solution

Answer

The solutions are \( x = 3 \) and \( x = -1 \).

Step 1: Understand the Equation

The given equation is \( 2^{x^2 - 2x} = 8 \). Notice that 8 can be rewritten as \( 2^3 \), making the equation \( 2^{x^2 - 2x} = 2^3 \).

Step 2: Set the Exponents Equal

Since the bases (2) on both sides of the equation are the same, we can set the exponents equal to each other: \( x^2 - 2x = 3 \).

Step 3: Rearrange the Equation

Bring all terms to one side of the equation: \( x^2 - 2x - 3 = 0 \). This is a standard quadratic equation.

Step 4: Solve the Quadratic Equation

Factor the quadratic equation: \( (x - 3)(x + 1) = 0 \).

Step 5: Find the Solutions

Set each factor equal to zero: \( x - 3 = 0 \) and \( x + 1 = 0 \). Solving these gives the solutions \( x = 3 \) and \( x = -1 \).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that involve repeated multiplication. For example, if you have a base number, like 2, raised to an exponent, this means multiplying the base by itself a certain number of times. Let's take a closer look at the problem "\(2^{x^2 - 2x} = 8\)." In this case, the base is 2, and the exponent is \(x^2 - 2x\). Notably, understanding how to manipulate exponents is crucial when both sides of an equation share the same base.

• When equations have the same base, their exponents can be set equal to each other.
• 8 can be expressed as \(2^3\), which is a key step in simplifying the equation.
• Thus, we have \(2^{x^2 - 2x} = 2^3\), allowing us to equate the exponents: \(x^2 - 2x = 3\).

By equating the exponents, complex exponential equations can be transformed into more manageable polynomial equations, setting the stage for the next steps like factoring.

Factoring Quadratics

Factoring quadratics is an essential skill in algebra, especially when dealing with quadratic equations, which are of the form \(ax^2 + bx + c = 0\). Factoring is the process of breaking down a quadratic into the product of simpler expressions, which can then be used to find the variable's solutions.Consider the quadratic equation \(x^2 - 2x - 3 = 0\) derived from equating the exponents. To factor, we look for two numbers that multiply to \(-3\) (the constant term) and add up to \(-2\) (the coefficient of \(x\)).

• The numbers \(-3\) and \(1\) fit these criteria because \(-3 \times 1 = -3\) and \(-3 + 1 = -2\).
• Thus, the equation can be factored into \((x - 3)(x + 1) = 0\).

Factoring quadratics simplifies solving, as it allows us to break down the equation into simpler parts, making finding solutions straightforward.

Solving Equations

Solving equations is all about finding the value(s) of the variable(s) that make the equation true. In the final step of our quadratic problem, after factoring \((x - 3)(x + 1) = 0\), we solve the equation by setting each factor equal to zero.This is based on the zero-product property, which states that if the product of two terms is zero, at least one of the terms must be zero.

• First, set \(x - 3 = 0\). By adding 3 to both sides, we find that \(x = 3\).
• Then, set \(x + 1 = 0\). Subtracting 1 from both sides gives us \(x = -1\).

Thus, the equation \(x^2 - 2x - 3 = 0\) has solutions \(x = 3\) and \(x = -1\). This straightforward approach underlines the power of factoring quadratics, as it simplifies finding solutions to polynomial equations.