Solving 18^(x^2 + 4x + 4) = 18^(9x + 18) A Step-by-Step Guide

Exponential equations are a fascinating area of mathematics, often appearing in various fields like physics, engineering, and finance. These equations involve variables in the exponents, requiring specific techniques to solve them. This article provides a comprehensive guide to solving the equation $18^{x^2 + 4x + 4} = 18^{9x + 18}$, delving into the underlying principles and step-by-step solutions. Understanding how to tackle such problems not only enhances your mathematical skills but also provides a foundation for more complex mathematical concepts. This exploration aims to provide a clear and detailed methodology, ensuring a solid grasp of the process.

Before we dive into the solution, it's crucial to understand the basic principles of exponential equations. An exponential equation is one in which the variable appears in the exponent. The general form can be expressed as $a^{f(x)} = a^{g(x)}$, where 'a' is the base and 'f(x)' and 'g(x)' are the exponents. A fundamental property of exponential equations is that if the bases are the same, then the exponents must be equal. This property forms the cornerstone of solving many exponential equations. In simpler terms, if $a^m = a^n$, then $m = n$. This is because the exponential function is a one-to-one function, meaning each input corresponds to a unique output. Understanding this principle allows us to transform complex exponential problems into simpler algebraic equations, making them easier to solve. Recognizing the structure of exponential equations and applying this property is the first step towards effectively solving them. For instance, in our problem, the base is 18 on both sides, allowing us to equate the exponents and form a quadratic equation, which we can then solve using standard algebraic techniques. This fundamental concept is not just limited to equations with numerical bases; it extends to cases involving functions and variables as well, making it a universally applicable tool in mathematics.

Our specific exponential equation is $18^{x^2 + 4x + 4} = 18^{9x + 18}$. This equation presents a clear example of two exponential expressions with the same base. Our goal is to find the values of 'x' that satisfy this equation. The first step in solving this equation is to recognize the common base, which is 18. Since the bases are the same, we can apply the principle that the exponents must be equal. This principle transforms the problem from an exponential equation into a more manageable algebraic equation. By equating the exponents, we set the stage for solving for 'x', which involves manipulating the resulting equation and finding its roots. This transformation is a critical step in simplifying the problem and making it solvable. It's also important to note that exponential equations of this form often lead to polynomial equations, such as quadratic equations, which have well-established methods for finding solutions. Therefore, understanding this initial transformation is key to successfully solving exponential equations.

Step 1: Equate the Exponents

As we've established, the first step in solving the exponential equation $18^{x^2 + 4x + 4} = 18^{9x + 18}$ is to equate the exponents. This is based on the fundamental property that if $a^m = a^n$, then $m = n$. Applying this to our equation, we set the exponent on the left side equal to the exponent on the right side: $x^2 + 4x + 4 = 9x + 18$. This step transforms the exponential equation into a quadratic equation, which is a polynomial equation of degree two. Quadratic equations are well-studied in algebra, and there are several methods to solve them, including factoring, completing the square, and using the quadratic formula. The transformation from exponential to quadratic is a crucial simplification, making the problem more accessible. This initial step is not only mathematically sound but also intuitive, as it reduces the complexity of the equation by focusing on the exponents themselves. This approach is a standard technique in solving exponential equations, and mastering it is essential for tackling more complex problems.

Step 2: Rearrange the Equation

Now that we have equated the exponents, our next step is to rearrange the quadratic equation $x^2 + 4x + 4 = 9x + 18$ into the standard quadratic form, which is $ax^2 + bx + c = 0$. This form is essential for applying various methods to solve quadratic equations, such as factoring or using the quadratic formula. To rearrange the equation, we need to move all terms to one side, leaving zero on the other side. Subtracting $9x$ and $18$ from both sides of the equation gives us: $x^2 + 4x + 4 - 9x - 18 = 0$. Now, we simplify by combining like terms. This involves adding or subtracting the coefficients of terms with the same power of 'x'. Combining the 'x' terms ($4x$ and $-9x$) gives us $-5x$, and combining the constant terms ($4$ and $-18$) gives us $-14$. Thus, the rearranged equation becomes $x^2 - 5x - 14 = 0$. This standard form of the quadratic equation is crucial because it allows us to easily identify the coefficients a, b, and c, which are necessary for applying the quadratic formula or attempting to factor the equation. The rearrangement step is a fundamental algebraic manipulation, and it's important to perform it accurately to avoid errors in the subsequent steps.

Step 3: Solve the Quadratic Equation

With the quadratic equation in standard form, $x^2 - 5x - 14 = 0$, we can now proceed to solve for 'x'. There are several methods to solve quadratic equations, but we will focus on factoring and the quadratic formula in this explanation. Factoring involves expressing the quadratic equation as a product of two binomials. To factor $x^2 - 5x - 14 = 0$, we look for two numbers that multiply to -14 and add up to -5. These numbers are -7 and 2. Therefore, we can factor the equation as $(x - 7)(x + 2) = 0$. Setting each factor equal to zero gives us two possible solutions: $x - 7 = 0$ and $x + 2 = 0$. Solving these equations yields $x = 7$ and $x = -2$. Alternatively, if factoring is not straightforward, we can use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the coefficients of the quadratic equation. In our case, $a = 1$, $b = -5$, and $c = -14$. Plugging these values into the quadratic formula, we get: $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)} = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2}$. This gives us two solutions: $x = \frac{5 + 9}{2} = 7$ and $x = \frac{5 - 9}{2} = -2$. Both methods, factoring and the quadratic formula, lead to the same solutions, which are the roots of the quadratic equation. The ability to solve quadratic equations is a fundamental skill in algebra, and it's crucial for solving many types of mathematical problems.

Step 4: Verify the Solutions

After finding the solutions for 'x', it's always a good practice to verify them by substituting them back into the original equation. This step ensures that the solutions are correct and that no errors were made during the solving process. Our original equation is $18^{x^2 + 4x + 4} = 18^{9x + 18}$, and our solutions are $x = 7$ and $x = -2$. Let's first verify $x = 7$. Substituting $x = 7$ into the equation gives us: $18^{(7^2 + 4(7) + 4)} = 18^{(49 + 28 + 4)} = 18^{81}$ on the left side and $18^{(9(7) + 18)} = 18^{(63 + 18)} = 18^{81}$ on the right side. Since both sides are equal, $x = 7$ is a valid solution. Now, let's verify $x = -2$. Substituting $x = -2$ into the equation gives us: $18^{((-2)^2 + 4(-2) + 4)} = 18^{(4 - 8 + 4)} = 18^0$ on the left side and $18^{(9(-2) + 18)} = 18^{(-18 + 18)} = 18^0$ on the right side. Again, both sides are equal, so $x = -2$ is also a valid solution. Verification is a critical step, especially in exams or problem-solving scenarios, as it provides confidence in the correctness of the answers. It also helps in identifying any algebraic errors that might have occurred during the solution process. In summary, verifying the solutions ensures that they satisfy the original equation and are indeed the correct answers.

After solving the exponential equation $18^{x^2 + 4x + 4} = 18^{9x + 18}$ and verifying our solutions, we can confidently state the final answer. We found two solutions for 'x': $x = 7$ and $x = -2$. These values satisfy the original equation, as we confirmed in the verification step. Therefore, the solution set for the equation is $\{-2, 7\}$. This complete solution demonstrates the step-by-step process of solving exponential equations, from equating the exponents to solving the resulting quadratic equation and verifying the solutions. The problem illustrates the importance of understanding the properties of exponential functions and applying algebraic techniques to solve mathematical problems. Mastering these skills is crucial for success in mathematics and related fields. The final answer provides a clear and concise resolution to the problem, solidifying the understanding of the solution process. In conclusion, the values $x = 7$ and $x = -2$ are the solutions to the given exponential equation.

In this detailed guide, we have successfully solved the exponential equation $18^{x^2 + 4x + 4} = 18^{9x + 18}$. We started by understanding the fundamental principles of exponential equations, particularly the property that if the bases are the same, the exponents must be equal. This principle allowed us to transform the exponential equation into a quadratic equation. We then rearranged the quadratic equation into its standard form and solved it using factoring and the quadratic formula, obtaining the solutions $x = 7$ and $x = -2$. Finally, we verified these solutions by substituting them back into the original equation, ensuring their correctness. This step-by-step approach not only provides the solution to this specific problem but also illustrates a general method for solving exponential equations. The ability to solve such equations is a valuable skill in mathematics and has applications in various fields. The process involves a combination of exponential properties, algebraic manipulations, and problem-solving techniques. By mastering these concepts, one can tackle a wide range of mathematical challenges. The conclusion reinforces the importance of each step in the solution process and highlights the broader applicability of the techniques learned.