Solving $x^2 - 11x + 30 = 0$ A Comprehensive Guide

Jul 9, 2025 by qnaunigon 51 views

Solving Quadratic Equations A Comprehensive Guide to $x^2 - 11x + 30 = 0$

Introduction

In the realm of mathematics, solving equations forms a cornerstone of various disciplines, from algebra to calculus and beyond. Quadratic equations, characterized by the presence of a squared term, hold a special significance due to their frequent appearance in real-world applications. In this article, we delve into the intricacies of solving the quadratic equation $x^2 - 11x + 30 = 0$ . Our exploration will cover the fundamental concepts, diverse solution methods, and practical applications, providing a comprehensive understanding of this essential mathematical concept.

Understanding Quadratic Equations

At its core, a quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is expressed as $ax^2 + bx + c = 0$ , where 'a', 'b', and 'c' are constants, with 'a' not equal to zero. The solutions to a quadratic equation, also known as roots or zeros, represent the values of 'x' that satisfy the equation. These roots correspond to the points where the parabola defined by the quadratic equation intersects the x-axis.

In the specific equation $x^2 - 11x + 30 = 0$ , we can identify the coefficients as follows: a = 1, b = -11, and c = 30. The goal is to find the values of 'x' that make this equation true. These values represent the points where the parabola represented by the equation crosses the x-axis on a graph. To find these solutions, we can employ a variety of methods, including factoring, completing the square, and the quadratic formula. Each approach offers a unique perspective and can be particularly useful depending on the specific characteristics of the equation.

Methods for Solving Quadratic Equations

1. Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. This method is particularly efficient when the equation can be factored easily. The key idea behind factoring is to reverse the process of expanding two binomials. We look for two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b). Once we've found these numbers, we can rewrite the quadratic equation in its factored form. Setting each factor equal to zero then gives us the solutions for 'x'.

For the equation $x^2 - 11x + 30 = 0$ , we seek two numbers that multiply to 30 and add up to -11. After some consideration, we can identify the numbers -5 and -6 as fulfilling these conditions. Therefore, we can factor the equation as follows:

$(x - 5)(x - 6) = 0$

Setting each factor equal to zero, we get:

x - 5 = 0 or x - 6 = 0

Solving these equations, we find the solutions:

x = 5 or x = 6

Thus, the roots of the quadratic equation $x^2 - 11x + 30 = 0$ are 5 and 6. Factoring is often the quickest method when the roots are integers or simple fractions, making it a valuable tool in solving quadratic equations.

2. Completing the Square

Completing the square is a versatile method for solving quadratic equations, even when factoring is not straightforward. This technique transforms the equation into a perfect square trinomial, which can then be easily solved by taking the square root. The process involves manipulating the equation to create a squared binomial on one side and a constant term on the other. This allows us to isolate 'x' and find the solutions.

To complete the square for the equation $x^2 - 11x + 30 = 0$ , we follow these steps:

Move the constant term to the right side of the equation:

$x^2 - 11x = -30$
Take half of the coefficient of the 'x' term (-11), square it ((-11/2)^2 = 121/4), and add it to both sides of the equation:

$x^2 - 11x + 121/4 = -30 + 121/4$
Rewrite the left side as a perfect square:

$(x - 11/2)^2 = 1/4$
Take the square root of both sides:

$x - 11/2 = ±√(1/4)$
Solve for 'x':

$x = 11/2 ± 1/2$

This gives us two solutions:

$x = 11/2 + 1/2 = 6$

$x = 11/2 - 1/2 = 5$

Thus, completing the square provides us with the same solutions, x = 5 and x = 6. This method is particularly useful when the quadratic equation cannot be easily factored or when dealing with complex roots.

3. Quadratic Formula

The quadratic formula is a universal solution for any quadratic equation, regardless of its factorability. It provides a direct method to find the roots, making it an indispensable tool in algebra. The formula is derived by applying the method of completing the square to the general form of a quadratic equation, $ax^2 + bx + c = 0$ . The resulting formula expresses the solutions 'x' in terms of the coefficients 'a', 'b', and 'c'.

The quadratic formula is given by:

$x = [-b ± √(b^2 - 4ac)] / 2a$

For the equation $x^2 - 11x + 30 = 0$ , we have a = 1, b = -11, and c = 30. Substituting these values into the quadratic formula, we get:

$x = [11 ± √((-11)^2 - 4 * 1 * 30)] / (2 * 1)$

Simplifying the expression:

$x = [11 ± √(121 - 120)] / 2$

$x = [11 ± √1] / 2$

$x = [11 ± 1] / 2$

This gives us two solutions:

$x = (11 + 1) / 2 = 6$

$x = (11 - 1) / 2 = 5$

As we can see, the quadratic formula confirms our previous results, yielding the solutions x = 5 and x = 6. The quadratic formula is especially valuable when the roots are irrational or complex, where factoring and completing the square may be more challenging.

Applications of Quadratic Equations

Quadratic equations are not merely abstract mathematical concepts; they have a wide range of applications in various fields, including physics, engineering, and economics. Their ability to model parabolic trajectories, optimize designs, and analyze economic trends makes them indispensable tools for problem-solving and decision-making.

In physics, quadratic equations are used to describe projectile motion. For example, the height of a projectile thrown into the air can be modeled as a quadratic function of time. Solving the quadratic equation allows us to determine the maximum height reached by the projectile and the time it takes to hit the ground. This information is crucial in fields like ballistics and sports science, where understanding the trajectory of objects is essential.

Engineering also relies heavily on quadratic equations. In structural engineering, quadratic equations can help determine the optimal shape and dimensions of arches and bridges, ensuring stability and load-bearing capacity. In electrical engineering, they are used to analyze circuits and design filters. The ability to model and solve quadratic equations enables engineers to create efficient and safe designs across various disciplines.

Economics utilizes quadratic equations to model supply and demand curves, cost functions, and revenue functions. These models help economists understand market behavior, predict economic trends, and optimize business strategies. For example, a quadratic equation might be used to find the price point that maximizes a company's profit. The insights gained from quadratic models can guide decision-making in areas such as pricing, production, and investment.

Conclusion

In conclusion, solving the quadratic equation $x^2 - 11x + 30 = 0$ provides a valuable illustration of the techniques and applications of quadratic equations. We explored three primary methods: factoring, completing the square, and the quadratic formula, each offering a unique approach to finding the solutions. Factoring is efficient for equations with easily identifiable factors, while completing the square provides a systematic method applicable to all quadratic equations. The quadratic formula serves as a universal tool, especially useful for equations with irrational or complex roots.

The roots of the equation $x^2 - 11x + 30 = 0$ are x = 5 and x = 6. These solutions represent the points where the parabola described by the equation intersects the x-axis. Beyond this specific example, we discussed the broad applicability of quadratic equations in diverse fields such as physics, engineering, and economics. From modeling projectile motion to optimizing engineering designs and analyzing economic trends, quadratic equations provide essential tools for problem-solving and decision-making.

Understanding quadratic equations is crucial for anyone pursuing studies in mathematics, science, or engineering. The ability to solve these equations effectively is a fundamental skill that opens doors to more advanced mathematical concepts and real-world applications. By mastering the techniques discussed in this article, you will be well-equipped to tackle a wide range of problems and appreciate the power and versatility of quadratic equations.