Solving (m-5)(m+6)=0 A Step By Step Guide To Quadratic Equations

by ADMIN 65 views

How to solve the quadratic equation (m-5)(m+6)=0?

In the realm of algebra, quadratic equations hold a prominent position, serving as fundamental tools for modeling and solving a myriad of real-world problems. These equations, characterized by their highest power of the variable being two, often present themselves in various forms, each requiring a specific approach to unravel their solutions. Among these forms, the factored form stands out as a particularly insightful representation, offering a direct pathway to the roots of the equation.

In this comprehensive guide, we embark on a step-by-step journey to demystify the process of solving a quadratic equation presented in factored form, specifically focusing on the equation (m-5)(m+6)=0. Through a clear and concise explanation of the underlying principles and techniques, we aim to empower readers with the confidence and skills to tackle similar problems with ease.

Understanding the Zero Product Property

At the heart of solving quadratic equations in factored form lies the Zero Product Property. This fundamental principle states that if the product of two or more factors equals zero, then at least one of the factors must be zero. In simpler terms, if we have an equation of the form A * B = 0, then either A = 0 or B = 0 (or both). This property serves as the cornerstone of our approach to solving the equation (m-5)(m+6)=0.

Applying the Zero Product Property to (m-5)(m+6)=0

Our equation, (m-5)(m+6)=0, is already conveniently presented in factored form. This means we have two factors, (m-5) and (m+6), whose product is equal to zero. According to the Zero Product Property, this implies that either (m-5) = 0 or (m+6) = 0 (or both).

Now, we have two simpler linear equations to solve. Let's tackle them one by one:

  1. Solving (m-5) = 0

    To isolate 'm', we add 5 to both sides of the equation:

    m - 5 + 5 = 0 + 5

    This simplifies to:

    m = 5

  2. Solving (m+6) = 0

    Similarly, to isolate 'm', we subtract 6 from both sides of the equation:

    m + 6 - 6 = 0 - 6

    This simplifies to:

    m = -6

The Solutions: m = 5 and m = -6

We have successfully determined the two solutions to the quadratic equation (m-5)(m+6)=0. These solutions, also known as roots or zeros of the equation, are m = 5 and m = -6. This means that when we substitute either 5 or -6 for 'm' in the original equation, the equation holds true.

Verifying the Solutions

To ensure the accuracy of our solutions, it's always a good practice to verify them by substituting them back into the original equation:

  1. Verifying m = 5

    Substituting m = 5 into (m-5)(m+6)=0, we get:

    (5-5)(5+6) = 0

    (0)(11) = 0

    0 = 0 (This confirms that m = 5 is a valid solution)

  2. Verifying m = -6

    Substituting m = -6 into (m-5)(m+6)=0, we get:

    (-6-5)(-6+6) = 0

    (-11)(0) = 0

    0 = 0 (This confirms that m = -6 is a valid solution)

Conclusion

In this comprehensive guide, we have meticulously walked through the process of solving the quadratic equation (m-5)(m+6)=0. By leveraging the Zero Product Property, we were able to transform the factored equation into two simpler linear equations, which we then solved to obtain the solutions m = 5 and m = -6. Furthermore, we verified these solutions to ensure their accuracy.

This step-by-step approach provides a solid foundation for tackling similar quadratic equations presented in factored form. By understanding the underlying principles and practicing these techniques, you can confidently navigate the world of quadratic equations and unlock their solutions.

Delving Deeper into Quadratic Equations and Their Applications

Now that we've mastered the art of solving quadratic equations in factored form, let's broaden our horizons and explore the broader context of quadratic equations and their diverse applications. Quadratic equations, with their characteristic form of ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero, are ubiquitous in mathematics, physics, engineering, and various other fields.

Beyond Factored Form: Other Solution Methods

While the factored form provides a direct route to solutions when applicable, not all quadratic equations readily present themselves in this format. In such cases, we turn to alternative methods, each with its own strengths and applications:

  • The Quadratic Formula: This versatile formula, derived from the process of completing the square, provides a guaranteed solution for any quadratic equation, regardless of its factorability. The formula is expressed as:

    x = (-b ± √(b^2 - 4ac)) / 2a

    Where 'a', 'b', and 'c' are the coefficients of the quadratic equation ax^2 + bx + c = 0.

  • Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial, which can then be factored and solved. While slightly more involved than the quadratic formula, completing the square offers valuable insights into the structure of quadratic equations and their relationship to perfect squares.

Real-World Applications of Quadratic Equations

Quadratic equations are not merely abstract mathematical constructs; they serve as powerful tools for modeling and solving a wide array of real-world problems. Here are just a few examples:

  • Projectile Motion: The trajectory of a projectile, such as a ball thrown into the air, can be accurately modeled using a quadratic equation. The equation takes into account the initial velocity, launch angle, and gravitational acceleration to predict the projectile's position at any given time.

  • Optimization Problems: Many optimization problems, where the goal is to maximize or minimize a certain quantity, can be formulated and solved using quadratic equations. For instance, determining the dimensions of a rectangular garden that maximizes the enclosed area for a given perimeter involves solving a quadratic equation.

  • Engineering Design: Quadratic equations play a crucial role in various engineering design applications, such as determining the optimal shape of a bridge arch or calculating the stress distribution in a structural component.

  • Financial Modeling: Quadratic equations can be used to model financial phenomena, such as the growth of investments or the depreciation of assets.

The Discriminant: Unveiling the Nature of Solutions

The discriminant, denoted as Δ (Delta), is a crucial component of the quadratic formula that provides valuable information about the nature of the solutions to a quadratic equation. The discriminant is defined as:

Δ = b^2 - 4ac

Where 'a', 'b', and 'c' are the coefficients of the quadratic equation ax^2 + bx + c = 0. The value of the discriminant reveals whether the quadratic equation has:

  • Two Distinct Real Solutions: If Δ > 0, the equation has two distinct real solutions.

  • One Real Solution (a Repeated Root): If Δ = 0, the equation has one real solution, which is a repeated root.

  • No Real Solutions (Two Complex Solutions): If Δ < 0, the equation has no real solutions, but it has two complex solutions.

Graphing Quadratic Equations: Parabolas

The graph of a quadratic equation is a parabola, a U-shaped curve that opens either upwards or downwards. The parabola's shape and position are determined by the coefficients of the quadratic equation. Key features of a parabola include:

  • Vertex: The vertex is the point where the parabola changes direction. It represents the minimum or maximum value of the quadratic function.

  • Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves.

  • X-intercepts (Roots): The x-intercepts are the points where the parabola intersects the x-axis. These points correspond to the real solutions of the quadratic equation.

  • Y-intercept: The y-intercept is the point where the parabola intersects the y-axis.

By understanding the relationship between quadratic equations and parabolas, we can gain a visual representation of the solutions and the behavior of the quadratic function.

Advanced Techniques and Applications of Quadratic Equations

Having established a strong foundation in the fundamentals of quadratic equations, let's venture into more advanced techniques and explore further applications that showcase the versatility of these mathematical tools.

Vieta's Formulas: Unveiling the Secrets of Roots

Vieta's formulas, named after the renowned French mathematician François Viète, provide a powerful connection between the coefficients of a polynomial equation and its roots. For a quadratic equation of the form ax^2 + bx + c = 0, Vieta's formulas state:

  • Sum of Roots: The sum of the roots (x1 + x2) is equal to -b/a.

  • Product of Roots: The product of the roots (x1 * x2) is equal to c/a.

Vieta's formulas offer a valuable shortcut for finding the sum and product of the roots without explicitly solving the quadratic equation. They also provide a means to check the validity of solutions obtained through other methods.

Quadratic Inequalities: Exploring the Realm of Inequalities

Quadratic inequalities extend the concept of quadratic equations by introducing inequality signs (>, <, ≥, ≤) into the expression. Solving quadratic inequalities involves finding the range of values for the variable that satisfy the inequality. The general approach involves:

  1. Transforming the inequality: Rewrite the inequality so that one side is zero.

  2. Finding the critical points: Determine the roots of the corresponding quadratic equation (where the expression equals zero).

  3. Creating a sign chart: Divide the number line into intervals based on the critical points and determine the sign of the quadratic expression in each interval.

  4. Identifying the solution set: Select the intervals that satisfy the original inequality.

Quadratic inequalities find applications in various fields, such as optimization problems, where we seek to find the range of values that maximize or minimize a certain function subject to constraints.

Systems of Equations Involving Quadratics

Systems of equations involving quadratics arise when we have two or more equations, at least one of which is quadratic, that need to be solved simultaneously. These systems can be solved using various methods, including:

  • Substitution: Solve one equation for one variable and substitute the expression into the other equation.

  • Elimination: Manipulate the equations to eliminate one variable and solve for the remaining variable.

  • Graphical Methods: Graph the equations and find the points of intersection, which represent the solutions to the system.

Systems of equations involving quadratics appear in diverse applications, such as determining the intersection points of curves, modeling the motion of objects, and solving optimization problems with multiple constraints.

Quadratic Forms: A Glimpse into Linear Algebra

Quadratic forms represent a generalization of quadratic equations to multiple variables. A quadratic form is a homogeneous polynomial of degree two in several variables. They play a significant role in linear algebra, optimization, and various other areas of mathematics and engineering.

Quadratic forms can be represented using matrices, and their properties can be analyzed using linear algebraic techniques. They find applications in areas such as:

  • Optimization: Finding the minimum or maximum of a quadratic form subject to constraints.

  • Statistics: Modeling the variance and covariance of random variables.

  • Engineering: Analyzing the stability of systems and designing control systems.

The Legacy of Quadratic Equations

In conclusion, quadratic equations stand as a cornerstone of mathematical knowledge, bridging the gap between abstract concepts and real-world applications. Their rich history, diverse solution methods, and far-reaching implications have solidified their place as an indispensable tool in the toolbox of mathematicians, scientists, engineers, and problem-solvers across various disciplines. From modeling projectile motion to optimizing financial portfolios, quadratic equations empower us to understand and shape the world around us.

As we continue to explore the vast landscape of mathematics, the principles and techniques associated with quadratic equations will undoubtedly serve as a solid foundation for further discoveries and advancements. The journey through quadratic equations is not merely an exercise in algebraic manipulation; it is an exploration of the power of mathematical thinking to unlock the secrets of the universe.