Solving $x^2-8x-20=0$ Using The Quadratic Formula A Step-by-Step Guide

In the realm of mathematics, quadratic equations hold a significant position. These equations, characterized by their highest power of 2, frequently arise in various scientific and engineering applications. Mastering the art of solving quadratic equations is, therefore, a fundamental skill for anyone delving into these fields. Among the methods available, the quadratic formula stands out as a universally applicable technique, capable of tackling any quadratic equation, regardless of its complexity. This article delves into the intricacies of using the quadratic formula, providing a step-by-step guide to solve the equation $x^2 - 8x - 20 = 0$. We will explore the formula itself, its derivation, and how to apply it effectively to find the solution set for the given equation. Understanding the quadratic formula not only equips you with a powerful problem-solving tool but also enhances your comprehension of the underlying mathematical principles governing these equations. By the end of this guide, you will be confident in your ability to tackle a wide range of quadratic equations using this versatile method. The journey into the world of quadratic equations begins with understanding their standard form and the significance of their coefficients, which play a crucial role in determining the nature and values of the solutions. So, let's embark on this mathematical exploration and unravel the power of the quadratic formula.

Understanding the Quadratic Formula

The quadratic formula is a cornerstone in algebra, a powerful tool designed to solve equations of the form $ax^2 + bx + c = 0$, where a, b, and c are constants, and a ≠ 0. This formula provides a direct method to find the values of x that satisfy the equation, commonly known as the roots or solutions. The beauty of the quadratic formula lies in its universality; it can solve any quadratic equation, regardless of whether the roots are real or complex, rational or irrational. It elegantly encapsulates the relationship between the coefficients of the quadratic equation and its solutions. The formula itself is expressed as:

$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

Each component of the formula plays a crucial role. The coefficients a, b, and c are directly substituted into the formula. The '±' symbol indicates that there are typically two solutions, one obtained by adding the square root term and the other by subtracting it. The term inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant is particularly important as it determines the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it is zero, the equation has exactly one real root (a repeated root); and if it is negative, the equation has two complex roots. Understanding the quadratic formula involves not only memorizing it but also grasping the significance of each term and its impact on the solutions. This understanding allows for a more intuitive approach to solving quadratic equations, paving the way for more advanced algebraic concepts. The formula is not just a mathematical trick; it is a consequence of completing the square, a fundamental algebraic technique, which provides a deeper insight into why the formula works.

Derivation of the Quadratic Formula

The derivation of the quadratic formula is a fascinating journey into the heart of algebraic manipulation. It stems from the method of completing the square, a technique used to rewrite a quadratic expression into a perfect square trinomial plus a constant. Starting with the standard form of a quadratic equation, $ax^2 + bx + c = 0$, where a ≠ 0, the derivation unfolds step by step. First, divide the entire equation by a to make the coefficient of $x^2$ equal to 1, resulting in:

$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

Next, move the constant term to the right side of the equation:

$x^2 + \frac{b}{a}x = -\frac{c}{a}$

Now comes the crucial step of completing the square. To do this, take half of the coefficient of the x term, which is $\frac{b}{2a}$, square it, which gives $(\frac{b}{2a})^2 = \frac{b^2}{4a^2}$, and add it to both sides of the equation. This ensures that the left side becomes a perfect square trinomial:

$x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}$

The left side can now be factored as a perfect square:

$(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$

Take the square root of both sides:

$x + \frac{b}{2a} = ±\frac{\sqrt{b^2 - 4ac}}{2a}$

Finally, isolate x by subtracting $\frac{b}{2a}$ from both sides:

$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

This elegant derivation not only provides the quadratic formula but also illuminates the underlying algebraic principles. Understanding this derivation enhances the appreciation for the formula's power and versatility. It underscores the importance of completing the square as a fundamental technique in algebra, applicable in various contexts beyond just deriving the quadratic formula. The derivation is a testament to the interconnectedness of mathematical concepts, showcasing how a seemingly complex formula can be built upon simpler, more intuitive ideas.

Applying the Quadratic Formula to $x^2 - 8x - 20 = 0$

Now, let's put the quadratic formula into action by solving the specific equation $x^2 - 8x - 20 = 0$. The first step is to identify the coefficients a, b, and c. In this equation, a = 1 (the coefficient of $x^2$), b = -8 (the coefficient of x), and c = -20 (the constant term). It's crucial to correctly identify these coefficients, as they will be directly substituted into the formula. Careless errors in this step can lead to incorrect solutions.

Next, substitute these values into the quadratic formula:

$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

$x = \frac{-(-8) ± \sqrt{(-8)^2 - 4(1)(-20)}}{2(1)}$

Now, simplify the expression step by step. First, simplify the terms inside the square root:

$x = \frac{8 ± \sqrt{64 + 80}}{2}$

$x = \frac{8 ± \sqrt{144}}{2}$

The square root of 144 is 12, so the equation becomes:

$x = \frac{8 ± 12}{2}$

This gives us two possible solutions, one with the '+' sign and one with the '-' sign.

For the '+' sign:

$x = \frac{8 + 12}{2} = \frac{20}{2} = 10$

For the '-' sign:

$x = \frac{8 - 12}{2} = \frac{-4}{2} = -2$

Thus, the two solutions for the equation $x^2 - 8x - 20 = 0$ are x = 10 and x = -2. These are the values of x that satisfy the original equation. To verify the solutions, you can substitute each value back into the original equation and check if the equation holds true. This step is a good practice to ensure the accuracy of your calculations. The process of applying the quadratic formula involves careful substitution and simplification, highlighting the importance of attention to detail in mathematical problem-solving. This example demonstrates the power and efficiency of the quadratic formula in finding the roots of a quadratic equation.

The Solution Set

Having found the two solutions for the equation $x^2 - 8x - 20 = 0$, which are x = 10 and x = -2, we can now express the solution set. The solution set is a collection of all the values that satisfy the equation, typically written within curly braces { }. In this case, the solution set is simply the set containing the two values we found.

Therefore, the solution set is {10, -2}.

This concise notation effectively communicates the complete set of solutions to the equation. It's important to note that the order in which the solutions are listed within the set does not matter. The set {-2, 10} is equivalent to {10, -2}. The solution set provides a clear and unambiguous answer to the problem, summarizing the results of our calculations. In the context of quadratic equations, the solution set can contain two real numbers, one real number (a repeated root), or two complex numbers. The nature of the solutions is determined by the discriminant, as discussed earlier. Understanding the concept of a solution set is crucial in mathematics, as it provides a formal way to express the solutions to any equation or system of equations. It's a fundamental concept that extends beyond quadratic equations and is used throughout various branches of mathematics. The solution set represents the culmination of the problem-solving process, providing a definitive answer to the question posed.

Importance of the Discriminant

As mentioned earlier, the discriminant, denoted as $b^2 - 4ac$, plays a pivotal role in determining the nature of the roots of a quadratic equation. This seemingly simple expression provides a wealth of information about the solutions without actually solving the equation. By examining the value of the discriminant, we can predict whether the equation will have two distinct real roots, one real root (a repeated root), or two complex roots. This insight is invaluable in various mathematical contexts, allowing us to understand the behavior of quadratic functions and their graphs.

If $b^2 - 4ac > 0$, the equation has two distinct real roots. This means the parabola represented by the quadratic equation intersects the x-axis at two different points. These roots correspond to the x-intercepts of the parabola. The fact that the discriminant is positive indicates that there are two real values of x that make the quadratic expression equal to zero.

If $b^2 - 4ac = 0$, the equation has exactly one real root (a repeated root). In this case, the parabola touches the x-axis at only one point. This root is a repeated solution, meaning it appears twice. The vertex of the parabola lies on the x-axis when the discriminant is zero.

If $b^2 - 4ac < 0$, the equation has two complex roots. Complex roots involve imaginary numbers, which are multiples of the imaginary unit i, where $i^2 = -1$. In this scenario, the parabola does not intersect the x-axis. The roots are complex conjugates, meaning they have the form a + bi and a - bi, where a and b are real numbers. Understanding complex roots is essential in various areas of mathematics and physics, particularly in electrical engineering and quantum mechanics.

The discriminant, therefore, serves as a powerful diagnostic tool, providing a quick and efficient way to classify the solutions of a quadratic equation. Its importance extends beyond simply solving equations; it provides a deeper understanding of the underlying mathematical structure and the behavior of quadratic functions. By analyzing the discriminant, mathematicians and scientists can gain valuable insights into the properties of quadratic equations and their applications in various fields.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have a wide array of real-world applications. From physics and engineering to economics and computer science, quadratic equations appear in various contexts, modeling a variety of phenomena. Their ability to describe curved paths and optimize quantities makes them indispensable tools in many fields.

In physics, quadratic equations are used to describe projectile motion. The trajectory of a projectile, such as a ball thrown into the air, can be modeled using a quadratic equation. The equation takes into account factors like initial velocity, launch angle, and gravitational acceleration. By solving the quadratic equation, physicists can determine the range, maximum height, and time of flight of the projectile. This understanding is crucial in sports, ballistics, and other areas involving motion under gravity.

Engineering also heavily relies on quadratic equations. In structural engineering, they are used to calculate the stresses and strains in beams and arches. The shape of a parabolic arch, for example, is described by a quadratic equation. By analyzing these equations, engineers can design structures that are strong and stable. In electrical engineering, quadratic equations are used to analyze circuits and calculate impedance, power, and other electrical quantities.

Economics utilizes quadratic equations to model cost, revenue, and profit functions. For example, the profit function of a company can often be represented by a quadratic equation. By finding the maximum value of the profit function (which corresponds to the vertex of the parabola), economists can determine the production level that maximizes profit. Similarly, quadratic equations can be used to model supply and demand curves, helping economists understand market dynamics.

Computer science also benefits from the application of quadratic equations. In computer graphics, they are used to generate curves and surfaces. Bezier curves, which are widely used in computer-aided design (CAD) and animation, are based on quadratic and cubic equations. These equations allow for the creation of smooth, aesthetically pleasing curves. In optimization algorithms, quadratic programming is a technique used to solve optimization problems where the objective function is quadratic and the constraints are linear. This has applications in machine learning, finance, and other areas.

The versatility of quadratic equations stems from their ability to model a wide range of phenomena characterized by a parabolic relationship. Whether it's the trajectory of a projectile, the shape of an arch, or the profit function of a company, quadratic equations provide a powerful tool for analysis and prediction. Their real-world applications underscore the importance of understanding and mastering these equations.

Conclusion

In conclusion, solving the equation $x^2 - 8x - 20 = 0$ using the quadratic formula demonstrates the power and versatility of this fundamental algebraic tool. We have walked through the steps of identifying the coefficients, substituting them into the formula, simplifying the expression, and arriving at the solutions x = 10 and x = -2. The solution set, {10, -2}, concisely represents the complete set of values that satisfy the equation. Beyond just finding the solutions, we have also explored the derivation of the quadratic formula, highlighting its connection to the method of completing the square. Understanding this derivation deepens our appreciation for the formula and its underlying mathematical principles.

Furthermore, we have discussed the significance of the discriminant, $b^2 - 4ac$, which provides valuable information about the nature of the roots without actually solving the equation. The discriminant allows us to predict whether the equation will have two distinct real roots, one real root (a repeated root), or two complex roots. This knowledge is crucial in various mathematical contexts and applications.

Finally, we have touched upon the real-world applications of quadratic equations, demonstrating their relevance in fields such as physics, engineering, economics, and computer science. From modeling projectile motion to optimizing profit functions, quadratic equations play a vital role in understanding and predicting a wide range of phenomena.

The quadratic formula is more than just a formula; it's a gateway to understanding a wide range of mathematical concepts and their applications. Mastering this formula and the principles behind it is an essential step in developing a strong foundation in algebra and mathematics. By understanding the quadratic formula, we empower ourselves to solve a wide range of problems and gain insights into the world around us.

By mastering the quadratic formula and understanding its applications, you gain a powerful tool for problem-solving in mathematics and beyond.