Solving The Quadratic Equation 16x² - 8x + 1 = 0 A Detailed Explanation
Understanding Quadratic Equations and Their Solutions
At the heart of algebra lies the elegant world of quadratic equations, equations that paint parabolas on the coordinate plane and hold the secrets to countless real-world phenomena. These equations, characterized by their squared variable term (x²), can have a fascinating array of solutions, and deciphering these solutions is a core skill in mathematics. When faced with a quadratic equation, we embark on a quest to discover the values of the variable (typically 'x') that make the equation a harmonious truth. These values are the equation's roots, the points where the parabola gracefully intersects the x-axis. The solution set is the collection of all these roots, a concise representation of the equation's soul. There are several avenues to explore when solving a quadratic equation. Factoring, the quadratic formula, and completing the square are among the most powerful tools in our arsenal. Each method has its strengths, and the optimal approach often depends on the specific equation at hand. Factoring, when applicable, offers a swift and elegant path to the solution. The quadratic formula, a universal key, unlocks the roots of any quadratic equation, regardless of its complexity. Completing the square, a technique that transforms the equation into a perfect square trinomial, provides both a solution and a deeper understanding of the equation's structure. In this article, we'll delve into the specific equation 16x² - 8x + 1 = 0, and together we'll unravel its solution set, exploring the methods, the subtleties, and the profound meaning behind the answer. We will explore the different possibilities for the solutions of a quadratic equation, which include two distinct real solutions, one repeated real solution, or two complex solutions. The nature of the solutions is dictated by the discriminant, a value derived from the coefficients of the quadratic equation. Through careful analysis, we will determine the exact nature of the solutions for our equation, revealing the hidden patterns within its algebraic form. So, let's embark on this mathematical journey, and together we'll illuminate the solution set of 16x² - 8x + 1 = 0, a testament to the beauty and power of algebra.
Exploring the Equation: 16x² - 8x + 1 = 0
Factoring the Quadratic Expression
In our quest to solve the equation 16x² - 8x + 1 = 0, we first turn to the powerful technique of factoring. Factoring, in essence, is the art of reverse multiplication, decomposing a complex expression into a product of simpler ones. When we encounter a quadratic expression, factoring allows us to rewrite it as a product of two binomials, each containing a variable term and a constant. The beauty of factoring lies in its ability to transform a seemingly intractable equation into a pair of easily solvable linear equations. To factor the quadratic expression 16x² - 8x + 1, we seek two binomials that, when multiplied, yield the original expression. We begin by observing the coefficients of the terms: 16, -8, and 1. These coefficients hold the key to unlocking the factors. The leading coefficient, 16, suggests that the binomials might involve terms like 4x and 4x, or perhaps 2x and 8x. The constant term, 1, indicates that the constant terms in the binomials will likely be either 1 and 1, or -1 and -1. The middle term, -8x, provides the crucial hint. It suggests that the constant terms in the binomials should both be negative, as their product will contribute to the negative coefficient of the middle term. With these clues in mind, we can attempt to construct the factors. After some careful consideration, we arrive at a promising candidate: (4x - 1)(4x - 1). To verify our factorization, we can multiply the binomials using the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last). Multiplying (4x - 1) by (4x - 1), we obtain: (4x * 4x) + (4x * -1) + (-1 * 4x) + (-1 * -1) = 16x² - 4x - 4x + 1 = 16x² - 8x + 1. Indeed, our factorization is correct! The quadratic expression 16x² - 8x + 1 can be elegantly expressed as (4x - 1)(4x - 1), or, more concisely, as (4x - 1)². This reveals a crucial aspect of our equation: it's a perfect square trinomial, a special type of quadratic expression that factors into the square of a binomial. Recognizing this perfect square nature simplifies our solution process immensely. We have transformed the equation 16x² - 8x + 1 = 0 into the equivalent form (4x - 1)² = 0. Now, we stand poised to solve for x, drawing upon the fundamental principle that if the square of a quantity is zero, the quantity itself must be zero. The factored form of our equation has unveiled a clear path to the solution, allowing us to isolate x and determine its value.
Determining the Solution Set
Solving for x
Having successfully factored the equation 16x² - 8x + 1 = 0 into the form (4x - 1)² = 0, we now stand on the precipice of determining the solution set. The factored form provides a direct route to the value(s) of x that satisfy the equation. The core principle we invoke is that if the square of a quantity is equal to zero, then the quantity itself must be zero. In mathematical terms, if a² = 0, then a = 0. Applying this principle to our equation, (4x - 1)² = 0, we can confidently assert that 4x - 1 = 0. We have transformed the original quadratic equation into a simpler linear equation, one that can be solved with ease. To isolate x, we perform a series of algebraic manipulations. First, we add 1 to both sides of the equation: 4x - 1 + 1 = 0 + 1, which simplifies to 4x = 1. Next, we divide both sides of the equation by 4: (4x) / 4 = 1 / 4, resulting in x = 1/4. This elegant solution, x = 1/4, is a root of our quadratic equation. It is the value of x that, when substituted back into the original equation, makes the equation a true statement. However, we must consider the nature of the solution. Because our factored form is (4x - 1)², the factor (4x - 1) appears twice. This indicates that the root x = 1/4 is a repeated root or a root with a multiplicity of 2. A repeated root signifies that the parabola represented by the quadratic equation touches the x-axis at only one point, rather than crossing it at two distinct points. The solution set, therefore, contains only one distinct value: 1/4. However, we acknowledge its multiplicity by stating that it is a repeated root. The concept of repeated roots is crucial in understanding the complete solution picture of a quadratic equation. It highlights the subtle ways in which a parabola can interact with the x-axis, revealing the nuances of algebraic solutions. In our case, the repeated root x = 1/4 signifies that the parabola of 16x² - 8x + 1 = 0 kisses the x-axis at the point (1/4, 0) without crossing it. This graphical interpretation provides a visual complement to our algebraic solution, solidifying our understanding of the equation's behavior. Therefore, the solution set of the equation 16x² - 8x + 1 = 0 is {1/4}, with the understanding that this root has a multiplicity of 2.
Describing the Solution Set
Interpreting the Results
Now that we have determined the solution set of the equation 16x² - 8x + 1 = 0, which consists of the single repeated root x = 1/4, we can accurately describe its nature. It's crucial to understand the implications of this solution within the context of quadratic equations. Recall that a quadratic equation, in its most general form, is ax² + bx + c = 0, where a, b, and c are constants. The solutions to a quadratic equation, also known as its roots, represent the x-intercepts of the parabola defined by the equation. These roots can be real numbers or complex numbers, and they can be distinct or repeated. The discriminant, denoted by Δ (Delta), plays a pivotal role in determining the nature of the roots. The discriminant is calculated using the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation. The value of the discriminant provides valuable information about the solutions: If Δ > 0, the equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points. If Δ = 0, the equation has one repeated real solution. In this case, the parabola touches the x-axis at a single point, its vertex. If Δ < 0, the equation has two complex solutions. This indicates that the parabola does not intersect the x-axis. For our specific equation, 16x² - 8x + 1 = 0, we have a = 16, b = -8, and c = 1. Let's calculate the discriminant: Δ = (-8)² - 4 * 16 * 1 = 64 - 64 = 0. The discriminant is zero, confirming our earlier finding that the equation has one repeated real solution. This aligns perfectly with our factored form, (4x - 1)² = 0, which clearly indicates a repeated root. The solution set {1/4} with multiplicity 2 signifies that the parabola represented by 16x² - 8x + 1 = 0 touches the x-axis only at the point (1/4, 0). The vertex of the parabola coincides with this point, and the parabola does not cross the x-axis. Therefore, the correct description of the solution set is that there is one real solution with a multiplicity of 2. This means that the quadratic expression has one repeated factor. In the context of the original options, this corresponds to the understanding that the expression factors into a perfect square. The repeated factor gives rise to the single, repeated solution. The discriminant provides a powerful tool for quickly assessing the nature of the solutions to a quadratic equation, saving us from the need to fully solve the equation in some cases. By calculating the discriminant, we gain valuable insights into the roots, their multiplicity, and the graphical behavior of the parabola.
Conclusion
Summarizing the Solution
In this comprehensive exploration, we have successfully unraveled the solution set of the quadratic equation 16x² - 8x + 1 = 0. We embarked on our journey by understanding the fundamental nature of quadratic equations and the significance of their solutions, or roots. We then delved into the powerful technique of factoring, which allowed us to transform the equation into the revealing form (4x - 1)² = 0. This factored form immediately illuminated the path to the solution: x = 1/4. However, our analysis didn't stop there. We recognized the crucial concept of repeated roots, noting that the factor (4x - 1) appears twice, indicating a multiplicity of 2 for the solution x = 1/4. This understanding led us to a precise description of the solution set: a single real solution, x = 1/4, with a multiplicity of 2. To further solidify our understanding, we invoked the discriminant, a key diagnostic tool for quadratic equations. By calculating the discriminant (Δ = b² - 4ac) for our equation, we obtained a value of 0. This confirmed our finding of a single repeated real solution, aligning perfectly with our factored form and our algebraic solution. The discriminant provided a concise and elegant way to characterize the nature of the roots without explicitly solving the equation. Our journey culminated in a clear and accurate description of the solution set, emphasizing the importance of considering both the value of the root and its multiplicity. The solution set {1/4} with multiplicity 2 signifies that the parabola represented by 16x² - 8x + 1 = 0 touches the x-axis at the point (1/4, 0), its vertex, and does not cross it. This graphical interpretation reinforces our algebraic findings, providing a holistic understanding of the equation's behavior. In essence, we have demonstrated the power of factoring, the significance of repeated roots, and the utility of the discriminant in solving and describing quadratic equations. We have also highlighted the interconnectedness of algebraic solutions and their graphical representations, showcasing the beauty and coherence of mathematics. The solution set of 16x² - 8x + 1 = 0 serves as a testament to the elegance and precision of algebraic methods, offering a glimpse into the profound world of quadratic equations and their diverse applications.
