Solving Quadratic Equations Using The Square Root Property
When dealing with quadratic equations, the square root property offers a straightforward method for finding solutions, especially when the equation can be expressed in a specific form. This article delves into the application of the square root property to solve the quadratic equation $81x^2 - 180x + 100 = 0$. We will break down each step, ensuring a clear understanding of the process. This method proves particularly useful when the quadratic equation can be manipulated into a form where a squared expression is isolated on one side of the equation, making it easier to find the roots or solutions. Understanding and applying the square root property not only simplifies the process of solving certain quadratic equations but also enhances your overall problem-solving skills in algebra. So, let’s explore how this property works and how it can be effectively utilized to solve quadratic equations.
Understanding the Square Root Property
The square root property states that if $u^2 = c$, where u is an algebraic expression and c is a real number, then $u = \sqrt{c}$ or $u = -\sqrt{c}$. In simpler terms, if the square of something equals a number, then that something equals either the positive or negative square root of that number. This property is a direct consequence of the definition of the square root and is a fundamental tool in solving equations where a variable is squared. The beauty of the square root property lies in its ability to directly address equations where a squared term is isolated, thereby avoiding the more complex procedures sometimes required by other methods, such as factoring or using the quadratic formula. The square root property is not just a mathematical shortcut; it's a concept deeply rooted in the principles of algebra, providing a clear and concise way to unravel equations and discover the values that satisfy them. By grasping this property, students and practitioners alike gain a more intuitive understanding of how to manipulate equations and arrive at solutions efficiently.
Applying the Square Root Property: A Step-by-Step Solution
To solve the quadratic equation $81x^2 - 180x + 100 = 0$, we can use the square root property. This involves several key steps, starting with recognizing and utilizing perfect square trinomials. This approach not only simplifies the equation but also allows for a more direct application of the square root property. By meticulously following each step, we can transform a seemingly complex equation into a manageable form, ultimately leading to the solutions. This method showcases the power of algebraic manipulation and the elegance of mathematical problem-solving. Let's proceed step-by-step to see how this property helps us unravel the mysteries of this quadratic equation and find its roots.
1. Recognize the Perfect Square Trinomial
The expression $81x^2 - 180x + 100$ is a perfect square trinomial. This means it can be factored into the form $(ax - b)^2$. Recognizing this pattern is crucial because it allows us to simplify the equation and apply the square root property more effectively. Perfect square trinomials have a distinct structure: the first term is a perfect square, the last term is a perfect square, and the middle term is twice the product of the square roots of the first and last terms. In this case, $81x^2$ is the square of $9x$, and $100$ is the square of $10$. The middle term, $-180x$, is indeed twice the product of $9x$ and $-10$, confirming that we have a perfect square trinomial. This recognition is not just a matter of pattern matching; it's a deep dive into the structure of algebraic expressions, revealing how different parts interact to form a cohesive whole. By mastering the identification of perfect square trinomials, you unlock a powerful tool in your algebraic arsenal, capable of transforming complex expressions into simpler, more manageable forms.
2. Factor the Trinomial
The perfect square trinomial $81x^2 - 180x + 100$ can be factored as $(9x - 10)^2$. This factorization is a direct result of recognizing the perfect square trinomial pattern, where the trinomial collapses into the square of a binomial. Factoring is a fundamental skill in algebra, allowing us to rewrite expressions in more simplified and insightful forms. In this case, the factorization not only simplifies the equation but also sets the stage for the application of the square root property. The process of factoring is akin to reverse multiplication, where we break down a complex expression into its constituent parts, revealing the underlying structure. By mastering factoring techniques, you gain a deeper understanding of algebraic relationships and enhance your ability to manipulate equations and solve problems effectively. This step is not just about finding the right factors; it's about seeing the elegance and interconnectedness of algebraic expressions.
3. Rewrite the Equation
Now, we can rewrite the equation $81x^2 - 180x + 100 = 0$ as $(9x - 10)^2 = 0$. This transformation is a crucial step in applying the square root property, as it isolates a squared expression on one side of the equation. By rewriting the equation in this form, we make it amenable to the direct application of the square root, which will allow us to solve for $x$. This step highlights the power of algebraic manipulation, where we transform equations into equivalent forms that are easier to solve. The ability to rewrite equations skillfully is a hallmark of mathematical proficiency, enabling us to tackle a wide range of problems with greater efficiency and insight. It's not just about changing the appearance of the equation; it's about revealing its underlying structure and setting the stage for a solution.
4. Apply the Square Root Property
Applying the square root property, we take the square root of both sides of the equation $(9x - 10)^2 = 0$. This gives us $9x - 10 = \pm \sqrt{0}$. Since the square root of 0 is 0, we have $9x - 10 = 0$. This step is the heart of the solution, where we directly address the squared expression and begin to isolate the variable $x$. The square root property allows us to undo the squaring operation, revealing the expression that was originally squared. This is a powerful technique in algebra, providing a direct pathway to solutions in equations where squared terms are present. It's not just a mechanical step; it's an application of a fundamental algebraic principle that connects squaring and square roots in a meaningful way.
5. Solve for x
To solve for x, we add 10 to both sides of the equation $9x - 10 = 0$, which yields $9x = 10$. Then, we divide both sides by 9 to get $x = \frac{10}{9}$. This final step isolates the variable $x$, giving us the solution to the quadratic equation. Solving for a variable is the ultimate goal in many algebraic problems, and this step demonstrates the culmination of our efforts in simplifying and manipulating the equation. It's not just about finding a numerical answer; it's about unraveling the relationship between the variable and the constants in the equation, revealing the value that satisfies the given condition. This process underscores the power of algebraic techniques in transforming equations and extracting solutions.
Final Answer
The solution to the quadratic equation $81x^2 - 180x + 100 = 0$ is $x = \frac{10}{9}$. This single solution arises from the unique nature of the perfect square trinomial, where the quadratic equation has a repeated root. The process of arriving at this solution involved recognizing and factoring the perfect square trinomial, applying the square root property, and isolating the variable $x$. This journey through the equation highlights the elegance and efficiency of algebraic techniques in solving problems. The final answer is not just a numerical value; it's the culmination of a logical and systematic process, demonstrating the power of mathematics in revealing the hidden relationships within equations. The satisfaction of finding the solution lies not only in the answer itself but also in the understanding of the steps and principles that led us there.
Therefore, the final answer is:
