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x2-11x+28=0: Mathematical Solution

x2-11x+28=0

In mathematics, x2-11x+28=0 quadratic equations occupy a significant and intriguing position. They manifest themselves in various domains, from physics and engineering to everyday problem-solving scenarios. Among these quadratic equations, x2-11x+28=0 is a noteworthy one. This article aims to investigate the intricacies of this equation deeply, elucidating its roots, factors, and the methodologies employed for its resolution. So, if you’ve ever found yourself perplexed when encountering the equation x2-11x+28=0, fear not! Please continue reading to demystify its enigma.

Welcome to exploring quadratic equations, an integral part of algebra and mathematics. This article will delve into quadratic equations, focusing on the equation x2-11x+28=0. Throughout this comprehensive discussion, we will uncover the general concept of quadratic equations, explore a range of methods to solve them, and conduct a deep dive into a specific case study to understand the practical applications of these equations.

What Constitutes a Quadratic Equation x2-11x+28=0?

Before we decipher the specifics of our equation, let’s take a moment to revisit the core concept of quadratic equations. A quadratic equation is a second-degree polynomial, typically represented in ax^2 + bx + c = 0. Here, ‘a,’ ‘b,’ and ‘c’ are constants, while ‘x’ is the variable we aim to solve for.

What Constitutes a Quadratic Equation x2-11x+28=0?

Understanding Quadratic Equations x2-11x+28=0

Before we dive into the specifics of the equation x^2 – 11x + 28 = 0, let’s first establish a foundational understanding of quadratic equations.

What is a Quadratic Equation?

To embark on this journey, we must first grasp the essence of a quadratic equation. A quadratic equation is a second-degree polynomial characterized by the squared variable. The most fundamental form of a quadratic equation is expressed as:

ax^2 + bx + c = 0

In this equation, ‘a,’ ‘b,’ and ‘c’ represent coefficients, while ‘x’ represents the variable. The equation x2-11x+28=0 precisely aligns with this standard form.

General Form of a Quadratic Equation

Understanding the general form of a quadratic equation is pivotal as it underpins the foundation of these equations. By recognizing this structure, we can conveniently identify the coefficients ‘a,’ ‘b,’ and ‘c.’ For our specific equation, ‘a’ is 1, ‘b’ is -11, and ‘c’ is 28.

The Equation x^2 – 11x + 28 = 0

Let’s focus on the equation: x^2 – 11x + 28 = 0.

Factoring the Quadratic Equation

One way to solve quadratic equations is by factoring. In this case, we can factor the equation as follows:

x^2 – 11x + 28 = 0 (x – 4)(x – 7) = 0

Finding the Roots

To find the solutions (roots) of the equation, we set each factor equal to zero:

x – 4 = 0 => x = 4 x – 7 = 0 => x = 7

So, the roots of the equation x^2 – 11x + 28 = 0 are x = 4 and x = 7.

Graphical Representation

Let’s visualize the solutions by graphing the equation.

Real-Life Applications

Quadratic equations are not just theoretical concepts; they have practical applications in various fields.

Physics

In physics, quadratic equations are used to describe objects’ motion under the influence of gravity. The equation x^2 – 11x + 28 = 0 might represent the time it takes for an object to reach a certain height when thrown upwards.

Solving Quadratic Equation x2-11x+28=0

Solving Quadratic Equation x2-11x+28=0

Quadratic equations are mathematical problems that can be tackled using various methods. Let’s explore two prominent techniques:

Factoring

Factoring is a technique to solve quadratic reckonings that can be easily broken down. However, for equations as complex as x2-11x+28=0, factoring may not be the most straightforward approach.

Quadratic Formula

The quadratic formula is a versatile and universally applicable approach to solving quadratic equations. This formula, often called upon, is:

x = (-b ± √(b^2 – 4ac)) / 2a

With this formula, we obtain the roots of the equation, representing the values of ‘x’ that render the equation equal to zero.

x^2-11x+28=0: A Case Study

Intriguingly, we now turn our attention to the specific equation x2-11x+28=0. Our exploration will involve a step-by-step analysis using the quadratic formula.

Step 1: Identifying the Coefficients

To initiate this journey, we must first identify the coefficients. In the equation, ‘a’ is 1, ‘b’ is -11, and ‘c’ is 28.

Step 2: Applying the Quadratic Formula

We can apply the quadratic formula with the coefficients now in our grasp. The equation takes shape as follows:

  • x = (-(-11) ± √((-11)^2 – 4 * 1 * 28)) / (2 * 1)

In the pursuit of a solution, we simplify the equation, leading to:

  • x = (11 ± √(121 – 112)) / 2
  • x = (11 ± √9) / 2

Step 3: Finding the Roots

As the equation begins to crystallize, we uncover two potential solutions:

  • x = (11 + 3) / 2 = 7
  • x = (11 – 3) / 2 = 4

Thus, the roots of the equation x2-11x+28=0 are revealed as x = 7 and x = 4.

Factoring the Quadratic Equation of x2-11x+28=0

Factoring the Quadratic Equation of x2-11x+28=0

Now, let’s immerse ourselves in the intricate process of factoring the equation x2-11x+28=0.

Identify the Coefficients

  • A = 1
  • B = -11
  • C = 28

To commence the factoring process, it is imperative to identify the coefficients of the equation, which in our case are A = 1, B = -11, and C = 28.

Find Two Numbers

Factoring a quadratic equation necessitates discovering two numbers that when multiplied, yield the product ‘A * C’ (in this instance, 1 * 28) and simultaneously add up to ‘B’ (which, in our case, is -11). The numbers we seek are -7 and -4, for the product of (-7) * (-4) indeed equals 28, and their sum, (-7) + (-4), equivalently amounts to -11.

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Rewrite the Equation

With our numbers identified, we can now proceed to rewrite the equation:

x² – 7x – 4x + 28 = 0

Grouping and Factoring

An essential step in the factoring process involves grouping the terms appropriately, which sets the stage for factoring by grouping. The equation in its grouped form appears as follows:

x(x – 7) – 4(x – 7) = 0

Apply the Distributive Property

Our next step involves the application of the distributive property, facilitating the factorization of the equation by extracting the common term (x – 7):

(x – 7)(x – 4) = 0

Finding the Roots

With the equation successfully factored in, we can now uncover its roots. This is achieved by setting each factor equal to zero:

  1. x – 7 = 0
  2. x – 4 = 0

Solving for ‘x’

Now that we’ve isolated the two factors, we proceed to solve for ‘x’ in each of the equations:

  1. x = 7
  2. x = 4.

The Roots of the Quadratic Equation of x2-11x+28=0

The Roots of the Quadratic Equation of x2-11x+28=0

Now that we’ve solved both equations derived from the factored form, we have two possible values for x:

x = 7

x = 4

These values represent the roots or solutions to the quadratic equation x2 – 11x + 28 = 0. In other words, if we substitute these values back into the original equation, it will hold:

For x = 7:

7² – 11(7) + 28 = 49 – 77 + 28 = 0

For x = 4:

4² – 11(4) + 28 = 16 – 44 + 28 = 0

Both values of x satisfy the equation, making them the roots of the quadratic equation.

Significance of the Solutions of x2-11x+28=0

Significance of the Solutions of x2-11x+28=0

The solutions, x=7 and x=4, carry profound significance in the context of the original problem. In real-world applications, quadratic equations often model motion, geometry, and optimization scenarios. For instance, if the quadratic equation represents the height of an object thrown into the air, the solutions correspond to the time points when the thing reaches certain heights. Understanding and interpreting these solutions are essential in physics, engineering, and economics.

Graphical Representation

Visualizing the quadratic equation x2 – 11x + 28 = 0 adds another layer of understanding. When plotted on a Cartesian plane, the equation forms a parabola. The roots, x=7 and x=4, correspond to the points where the parabola intersects the x-axis. This graphical representation reinforces the algebraic solutions and provides an intuitive grasp of the equation’s behavior.

Relationship with Factoring

Factoring is an alternative method to solve quadratic equations, and it can provide insights into the nature of the solutions. In the case of x2 – 11x + 28 = 0, factoring involves finding two binomials whose product equals the original expression. The factored form is (x−7)(x−4)=0, revealing that x=7 and x=4 are indeed the roots. This connection between factoring and the quadratic formula showcases the elegance and versatility of mathematical techniques.

Applications in Real-world Scenarios

Quadratic equations manifest in various practical situations, including x2 – 11x + 28 = 0. Consider a business owner analyzing profit and loss or a biologist studying population growth. These scenarios can be modeled using quadratic equations, with the solutions offering valuable insights into critical points and trends. Solving such equations is a powerful tool in decision-making and problem-solving across diverse fields.

Conclusion

The quadratic equation x^2 – 11x + 28 = 0 is a fundamental mathematical concept with real-world applications. By factoring and solving it, we found that it has two roots: x = 4 and x = 7. Understanding quadratic equations is crucial in mathematics and various fields, making it a valuable skill for problem-solving. To sum up, delving into quadratic equations has been a fascinating journey.

In this article, we have unraveled the fundamental concepts surrounding quadratic equations and the methodology of solving them using the quadratic formula. We have witnessed the practical implications through a specific case study, namely x2-11x+28=0. Furthermore, we have explored the visual representation of quadratic equations via parabolic graphs and highlighted their real-world relevance in various applications.

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