Problem 27 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example 2 . $$3^{x^{2}+4 x}=\frac{1}{81}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -2\).

Step 1: Express 81 as a Power of 3

First, recognize that 81 is a power of 3. Specifically, we can express 81 as \(81 = 3^4\). Therefore, \(\frac{1}{81} = \frac{1}{3^4} = 3^{-4}\). Rewrite the equation with this expression: \(3^{x^2 + 4x} = 3^{-4}\).

Step 2: Set Exponents Equal

Now that the bases are the same (both are powers of 3), set the exponents equal to each other. This gives us the equation \(x^2 + 4x = -4\).

Step 3: Rearrange as a Quadratic Equation

Move all terms to one side to make it a standard quadratic equation: \(x^2 + 4x + 4 = 0\).

Step 4: Factor the Quadratic Equation

Factor the quadratic equation. Notice that \(x^2 + 4x + 4\) can be factored as \((x + 2)(x + 2) = 0\), or equivalently \((x + 2)^2 = 0\).

Step 5: Solve for x

Solve the equation \((x + 2)^2 = 0\). This implies \(x + 2 = 0\). Therefore, solve for \(x\) to get \(x = -2\).

Key Concepts

Quadratic Equations

Quadratic equations form the foundation for solving many types of algebraic problems. A quadratic equation is any equation that can be rearranged in the form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. When expressed in this form, it’s known as the standard form of a quadratic equation. Understanding the structure is key:

• \(x^2\) is the quadratic term showing the highest power of the variable.
• \(x\) is the linear term.
• \(c\) is the constant term.

Quadratic equations typically have two solutions, which might be real or complex numbers. The solutions can be found using several methods, including factoring, the quadratic formula, or completing the square. The choice of method often depends on the specific form and complexity of the equation.

Factoring

Factoring is a common method for solving quadratic equations, especially when the quadratic can easily be broken into simpler components. For a quadratic like \(x^2 + 4x + 4\), factoring involves writing it as a product of two binomials: \[(x + 2)(x + 2) = 0\] This step is crucial because it simplifies the equation and illuminates the potential solutions. Mathematically, what's happening is the quadratic is being broken down into two expressions that, when multiplied together, give the original quadratic expression. Here's a handy guide to help you know when factoring works well:

• All terms can be divided by the same number or variable.
• The equation simplifies into recognizable binomials.
• After factoring, the zero-product property can be applied, allowing each binomial to be solved individually.

When done correctly, factoring transforms a complex-looking quadratic equation into a simpler, more solvable form.

Solving Equations

Solving equations is a fundamental skill in algebra. This involves finding the values of variables that satisfy an equation. With our factored equation \((x + 2)(x + 2) = 0\), we utilize an essential principle: if the product of two factors is zero, then at least one of the factors must be zero. The zero-product property aids us in this:

• Set each factor equal to zero, such as \(x + 2 = 0\).
• Solve for the variable \(x\), giving you \(x = -2\).

This means the equation has a repeated root at \(x = -2\). This approach shows that by breaking down the equation, we can systematically find solutions even without technology. The steps involved in solving quadratic equations using factorization are not just algebraic manipulations but also foster logical thinking and problem-solving skills.