Problem 38 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(3^{x^{2}+2}=3^{6}\)

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -2\).

Step 1: Understand the Equation

The equation given is \(3^{x^2 + 2} = 3^6\). Since the bases are the same (both are 3), we can set the exponents equal to each other. This is based on the property of exponents which states that if \(a^m = a^n\), then \(m = n\).

Step 2: Set the Exponents Equal

Using the property from Step 1, set the exponents \(x^2 + 2\) equal to 6. So, the equation becomes \(x^2 + 2 = 6\).

Step 3: Solve for x

Subtract 2 from both sides of the equation \(x^2 + 2 = 6\) to get \(x^2 = 4\). Then, take the square root of both sides to solve for \(x\). The solutions are \(x = 2\) and \(x = -2\), since both positive and negative roots must be considered.

Step 4: Check the Solutions

To verify the solutions, substitute \(x = 2\) and \(x = -2\) back into the original equation. - For \(x = 2\), calculate \(3^{2^2 + 2} = 3^{4 + 2} = 3^6\), which matches exactly.- For \(x = -2\), calculate \(3^{(-2)^2 + 2} = 3^{4 + 2} = 3^6\), which also matches.Both solutions satisfy the original equation.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with exponential equations. In the original exercise, we used the property: if the bases of two powers are the same, the exponents must also be equal.
This leads to the rule: if \(a^m = a^n\), then \(m = n\).
For instance, with \(3^{x^2 + 2} = 3^6\), both sides have the base 3, so equating the exponents \(x^2 + 2 = 6\) helps simplify the task without diving into logarithms or other methods.
This property makes solving exponential equations more approachable, allowing you to reduce complex expressions to simpler algebraic forms. Consider the base consistency as the golden rule here, especially useful when exponents become more intricate. Recognizing common bases quickly and applying the equality of exponents is a time-saving skill in algebra.

Solving Quadratic Equations

Once the exponents were equated \(x^2 + 2 = 6\), the task becomes solving a basic quadratic equation. Quadratics are foundational in algebra because they appear often and have predictable solving methods.
Rearranging gives \(x^2 = 4\), a simple equation to handle. The next step is crucial: taking the square root of both sides.
Square roots have two potential results: positive and negative. Thus, \(x = \pm \sqrt{4}\) leads to \(x = 2\) and \(x = -2\).
Key methods for solving quadratics in more complex scenarios include:

• Factoring: breaking down equations into products of simpler expressions.
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for equations \(ax^2 + bx + c = 0\).
• Completing the square: transforming a quadratic into a perfect square trinomial.

Understanding these techniques will enable you to solve any quadratic equation you might encounter.

Checking Solutions in Algebra

Always verify your solutions once you solve equations, especially when they involve exponents or quadratics. Checking ensures you recognize mistakes early.
For our two solutions \(x = 2\) and \(x = -2\), substitute back into the original equation to verify correctness:

• Substitute \(x = 2\): - Calculate \(3^{2^2 + 2} = 3^{4 + 2} = 3^6\); correct since both sides equal \(3^6\).
• Substitute \(x = -2\): - Calculate \(3^{(-2)^2 + 2} = 3^{4 + 2} = 3^6\); correct, maintaining equation equality.

This practice confirms each solution is valid. Discovering any discrepancies here might suggest reviewing your arithmetic or understanding of earlier steps. In complex problems, checking ensures all possible solutions and interpretations of equations are considered.