Solving $(x-3)^2 = -4$ Using The Square Root Property

In the realm of mathematics, the square root property of equality stands as a vital tool for solving certain types of equations, particularly those involving squared expressions. This method provides a direct and efficient way to isolate the variable and determine the solutions. In this article, we delve into the application of the square root property of equality, illustrating its use with a specific example: solving the equation $(x-3)^2 = -4$. We will explore the underlying principles, demonstrate the step-by-step solution process, and discuss the implications of the results. Understanding the square root property is crucial for students and anyone involved in mathematical problem-solving, as it appears in various contexts, from basic algebra to more advanced calculus. The ability to confidently apply this property enhances one's mathematical toolkit and problem-solving skills. Furthermore, we will analyze the nature of the solutions obtained in this particular problem, which introduces the concept of imaginary numbers, adding another layer of complexity and insight into the mathematical landscape. So, let's embark on this journey of mathematical exploration, unraveling the mysteries of the square root property and its applications.

Understanding the Square Root Property of Equality

The square root property of equality is a fundamental principle in algebra that states: if two quantities are equal, then their square roots are also equal. More formally, if $a^2 = b$, then $a = ±√b$. This property stems directly from the definition of the square root operation, which seeks a number that, when multiplied by itself, yields the original number. The “±” symbol is crucial because it indicates that there are typically two solutions: a positive square root and a negative square root. For instance, the square root of 9 is both 3 and -3, since 3² = 9 and (-3)² = 9. The square root property is especially useful when solving equations in the form of a squared expression equal to a constant, such as $(x-h)² = k$, where x is the variable, and h and k are constants. In such cases, applying the square root property directly allows us to eliminate the square, simplify the equation, and isolate x. This method avoids the need for expanding the squared expression and using other techniques like factoring or the quadratic formula, providing a more straightforward path to the solution. However, it’s important to note that the square root property is applicable only when the squared expression is isolated on one side of the equation. If there are additional terms or coefficients, they must be addressed before applying the square root property. Understanding this property is a cornerstone in algebra, enabling efficient and accurate problem-solving in a variety of mathematical contexts.

Solving the Equation $(x-3)^2 = -4$

To solve the equation $(x-3)^2 = -4$ using the square root property of equality, we will follow a systematic step-by-step approach. Our goal is to isolate $x$ and find all possible values that satisfy the equation. This process involves carefully applying the square root property, considering both positive and negative roots, and dealing with the implications of the constant on the right-hand side being negative. Each step is designed to bring us closer to the solution while maintaining the integrity of the equation. The first step is to recognize that the equation is in the form of a squared expression equal to a constant, which makes the square root property the ideal method to use. By taking the square root of both sides, we can eliminate the square on the left-hand side and begin the process of isolating $x$. This method provides a clear and efficient pathway to the solution, avoiding the complexities that other methods might entail. Let's begin by applying the square root property to both sides of the equation and then carefully address the resulting expression to determine the final solutions. This process will not only provide the answers but also demonstrate the power and elegance of the square root property in solving certain algebraic equations.

Step 1: Apply the Square Root Property

The initial step in solving the equation $(x-3)^2 = -4$ using the square root property is to take the square root of both sides. This step directly applies the principle that if two quantities are equal, their square roots are also equal. By taking the square root of both sides, we can effectively eliminate the square on the left-hand side of the equation, which contains the variable term. This allows us to simplify the equation and begin the process of isolating the variable $x$. When we apply the square root property, it’s crucial to remember to consider both the positive and negative square roots, as squaring either a positive or a negative number results in a positive number. Therefore, we introduce the “±” symbol to indicate that there are two potential roots. Applying this to our equation, we get: $√(x-3)^2 = ±√(-4)$. This step is the cornerstone of using the square root property and sets the stage for further simplification. It transforms the equation into a more manageable form, where we can focus on isolating $x$. This method provides a direct pathway to the solution, making it an efficient and effective technique for solving equations of this type. By carefully applying the square root property, we pave the way for the subsequent steps that will ultimately reveal the solutions to the equation.

Step 2: Simplify the Equation

Following the application of the square root property, the next step is to simplify the equation. We now have $√(x-3)^2 = ±√(-4)$. The square root of $(x-3)^2$ is simply $(x-3)$, because the square root operation effectively cancels out the square. On the right-hand side, we encounter the square root of a negative number, specifically $√(-4)$. This introduces the concept of imaginary numbers, as the square root of a negative number is not a real number. To simplify $√(-4)$, we can rewrite it as $√(4 * -1)$. We know that $√4 = 2$, and the square root of $-1$ is defined as the imaginary unit, denoted by $i$. Therefore, $√(-4)$ can be expressed as $2i$. Consequently, $±√(-4)$ becomes $±2i$. Incorporating these simplifications, our equation now reads: $x - 3 = ±2i$. This step is crucial because it transforms the equation from one involving a squared expression and a square root into a simpler linear form. By recognizing and correctly handling the imaginary unit, we maintain the mathematical integrity of the equation and pave the way for the final step of isolating $x$. This simplification is a key component in solving equations involving square roots of negative numbers and highlights the importance of understanding imaginary numbers in algebra.

Step 3: Isolate x

The final step in solving the equation $(x-3)^2 = -4$ is to isolate $x$. We have simplified the equation to $x - 3 = ±2i$. To isolate $x$, we need to eliminate the $-3$ on the left-hand side of the equation. This is achieved by adding $3$ to both sides of the equation, maintaining the equality. Performing this operation, we get: $x - 3 + 3 = 3 ± 2i$, which simplifies to $x = 3 ± 2i$. This result indicates that there are two solutions for $x$: one where we add $2i$ to $3$ and another where we subtract $2i$ from $3$. These solutions are complex numbers, each having a real part (3) and an imaginary part ($2i$ and $-2i$, respectively). The solutions can be written as $x = 3 + 2i$ and $x = 3 - 2i$. These are the values of $x$ that, when substituted back into the original equation, will satisfy the equation. The isolation of $x$ completes the solution process, providing us with the definitive answers to the equation. This final step demonstrates the elegance and precision of algebra in solving mathematical problems, showcasing how a series of carefully executed steps can lead to a clear and accurate solution.

The Solutions

The solutions to the equation $(x-3)^2 = -4$ are $x = 3 + 2i$ and $x = 3 - 2i$. These solutions are complex numbers, each consisting of a real part and an imaginary part. The real part is $3$, and the imaginary parts are $2i$ and $-2i$, respectively. Complex solutions arise because we encountered the square root of a negative number during the solution process. When we took the square root of $-4$, we obtained $±2i$, where $i$ represents the imaginary unit, defined as the square root of $-1$. These complex solutions indicate that the original equation has no real number solutions. In graphical terms, this means that the parabola represented by the equation $y = (x-3)^2 + 4$ does not intersect the x-axis. The complex solutions are a direct consequence of the negative constant on the right-hand side of the original equation. This highlights the importance of understanding complex numbers in algebra, as they appear when solving equations that do not have real number solutions. The ability to identify and interpret complex solutions is a crucial skill in advanced mathematics and is essential for a comprehensive understanding of equation solving. These solutions provide a complete answer to the problem, demonstrating the full range of possibilities when applying the square root property.

In conclusion, the square root property of equality is a powerful and efficient tool for solving equations of the form $(x-h)^2 = k$. By applying this property, we can directly address squared expressions and simplify equations, often avoiding more complex methods such as factoring or the quadratic formula. In the specific case of the equation $(x-3)^2 = -4$, we encountered a scenario where the solutions were complex numbers, $x = 3 + 2i$ and $x = 3 - 2i$. This underscores the importance of understanding complex numbers in algebra and recognizing when they arise as solutions. The step-by-step process we followed—applying the square root property, simplifying the equation, and isolating the variable—demonstrates the systematic approach needed for solving algebraic equations. The square root property not only provides a method for finding solutions but also enhances our understanding of the nature of those solutions, including the possibility of complex numbers. This mathematical tool is invaluable for students and professionals alike, providing a clear and concise way to tackle a specific class of equations. The ability to confidently apply the square root property contributes to a more robust mathematical skill set and a deeper appreciation for the elegance and power of algebraic techniques.