Find The Smallest Root Of The Equation (x² - 3x - 10)(7 - X) = 0

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Introduction: Mastering Quadratic Equations and Root Identification

In the realm of mathematics, solving equations is a fundamental skill that underpins numerous concepts and applications. Among the various types of equations, quadratic equations hold a prominent position due to their widespread occurrence in various fields, ranging from physics and engineering to economics and finance. This article delves into the intricacies of solving a specific equation, (x² - 3x - 10)(7 - x) = 0, with the primary objective of identifying its smallest root. By systematically dissecting the equation and employing established algebraic techniques, we will not only pinpoint the correct answer but also gain a deeper understanding of the underlying principles involved in solving such problems. This exploration will serve as a valuable resource for students, educators, and anyone seeking to enhance their mathematical prowess.

Deconstructing the Equation: A Step-by-Step Approach

The equation at hand, (x² - 3x - 10)(7 - x) = 0, presents a unique structure that allows us to leverage the zero-product property. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In our case, the equation is a product of two factors: (x² - 3x - 10) and (7 - x). Therefore, to find the roots of the equation, we need to set each factor equal to zero and solve for x.

Tackling the Quadratic Factor: x² - 3x - 10 = 0

The first factor, x² - 3x - 10, is a quadratic expression. To solve the quadratic equation x² - 3x - 10 = 0, we can employ several methods, including factoring, completing the square, or using the quadratic formula. In this instance, factoring proves to be the most efficient approach. We seek two numbers that multiply to -10 and add up to -3. These numbers are -5 and 2. Thus, we can factor the quadratic expression as follows:

x² - 3x - 10 = (x - 5)(x + 2) = 0

Setting each factor equal to zero, we obtain two potential roots:

x - 5 = 0 => x = 5

x + 2 = 0 => x = -2

Addressing the Linear Factor: 7 - x = 0

The second factor, 7 - x, is a linear expression. Setting it equal to zero, we have:

7 - x = 0 => x = 7

Identifying the Roots: A Comprehensive Overview

Having solved both factors, we have identified three potential roots for the equation (x² - 3x - 10)(7 - x) = 0: -2, 5, and 7. These roots represent the values of x that make the equation true. To ensure accuracy, we can substitute each root back into the original equation and verify that the result is indeed zero. This process serves as a crucial check, especially when dealing with more complex equations.

The Smallest Root: Unveiling the Solution

Our primary objective is to determine the smallest root among the identified solutions. By comparing the values -2, 5, and 7, it becomes evident that -2 is the smallest. Therefore, the smallest root of the equation (x² - 3x - 10)(7 - x) = 0 is -2. This concludes our step-by-step analysis and provides a definitive answer to the problem.

Answer the Question: The Smallest Root

The smallest root of the equation (x² - 3x - 10)(7 - x) = 0 is B. -2.

Exploring Alternative Approaches: A Broader Perspective

While factoring proved to be an efficient method for solving the quadratic equation in this case, it is essential to acknowledge that alternative approaches exist. Two prominent alternatives are:

  1. Completing the Square: This technique involves manipulating the quadratic expression to create a perfect square trinomial, which can then be easily solved. While slightly more involved than factoring, completing the square is a valuable method, especially when factoring is not straightforward.

  2. The Quadratic Formula: This formula provides a universal solution for any quadratic equation of the form ax² + bx + c = 0. The formula is given by:

    x = (-b ± √(b² - 4ac)) / 2a

    In our case, a = 1, b = -3, and c = -10. Substituting these values into the quadratic formula, we obtain the same roots as before: 5 and -2. The quadratic formula serves as a powerful tool when factoring or completing the square proves challenging.

Conceptual Understanding: The Importance of Roots

The roots of an equation hold significant importance in mathematics and its applications. They represent the values of the variable that satisfy the equation, effectively making it true. In the context of a polynomial equation, the roots correspond to the x-intercepts of the polynomial's graph. Understanding the concept of roots is crucial for solving equations, analyzing functions, and modeling real-world phenomena.

Real-World Applications: The Pervasiveness of Quadratic Equations

Quadratic equations are not merely abstract mathematical constructs; they manifest themselves in a multitude of real-world scenarios. From the trajectory of a projectile to the design of parabolic reflectors, quadratic equations play a vital role in various fields. Here are a few examples:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, such as the path of a ball thrown into the air.
  • Engineering: Quadratic equations are employed in the design of bridges, arches, and other structures, ensuring their stability and strength.
  • Finance: Quadratic equations can be used to model investment growth and calculate the optimal interest rates.
  • Computer Graphics: Quadratic equations are utilized in creating curves and surfaces in computer graphics and animations.

These examples illustrate the pervasive nature of quadratic equations and their significance in diverse domains. By mastering the techniques for solving these equations, we equip ourselves with valuable tools for tackling real-world problems.

Conclusion: A Journey Through Equation Solving

In this comprehensive exploration, we embarked on a journey to solve the equation (x² - 3x - 10)(7 - x) = 0 and identify its smallest root. By leveraging the zero-product property, factoring the quadratic expression, and solving the linear factor, we successfully determined that the smallest root is -2. Furthermore, we delved into alternative approaches, such as completing the square and the quadratic formula, and emphasized the conceptual understanding of roots and their real-world applications. This article serves as a testament to the power of systematic problem-solving and the importance of mathematical literacy in navigating the complexities of the world around us. By embracing these principles, we empower ourselves to tackle a wide range of challenges and contribute meaningfully to society.