x2-11x+28=0 : Solving the Quadratic Equation

When it comes to solving mathematical equations, few topics are as crucial for students as quadratic equations. Among these, equations like x² – 11x + 28 = 0 represent a standard problem that introduces students to the principles of algebra. Understanding how to solve quadratic equations is essential, not only for academic success but also for real-life applications in engineering, physics, economics, and beyond. In this comprehensive article, we will explore how to solve the quadratic equation x² – 11x + 28 = 0, analyze different methods of solving it, and discuss the importance of quadratic equations in mathematics.

Table of Contents

What is Quadratic Equation?

Quadratic equation is an algebraic equation of second degree in x only namely ax^2 + bx + c = 0 where a, b, c are constants and their values are fixed. The general format of the quadratic equation is ax² + bx + c = 0, and within this format and x is term as the variable while b and c are known as the constants of the variables and c as a constant factor. Another requirement that every quadratic equation (a ≠ 0) must possess is that, the coefficient of x2 must carry a non zero term. It is important to notice that the x2 term comes first and the x term comes second and the constant term follows when putting a quadratic equation in its standard forma = 1

b = -11

c = 28

Factoring the Quadratic Equation

Quadratic problems can be solved in part by factoring. The goal for x2-11x+28=0 quadratic formula
factorization into two binomials, which then multiply to yield the original equation. We are looking for two numbers that multiply to the constant term 28 and add up to the middle term’s coefficient of -11. We can now rewrite the equation using these values:

There are now two components left in the equation:

x – 7 = 0
x – 4 = 0

Solving Equation: x-7=0

To isolate x, add seven to either side:

x=7

To isolate x, multiply both sides by 4:

x=4

Roots of the Quadratic Equation

You will enter the value in the equation when you have obtained the value from the equation.

x = 7

x = 4

These values are the quadratic equation x2-11x+28=0 quadratic formula’s roots or solutions. Stated otherwise, the following will occur if we substitute these numbers back into the original equation:

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 serve as the quadratic equation’s roots and solve the problem.

Graphical Representation

You may see the equation x2-11x+28=0 quadratic formula factorizacion graphically here.

Plotting the equation: The equation x2-11x+28=0 quadratic formula can be graphed to see its features and form. When the plotting equation is applied on a coordinate plane, an ellipse with specific properties is generated.
Analyzing the graph: The graph representing x2-11x+28=0 quadratic formula provides important insights into the equation. The Ellipse’s centroid is situated at (0 and 0), and its major and minor axes can be found using the coefficients of x2 and y2. Symmetry along the x and y axes is also discernible because of the squared terms and the graph.

Applications of thе Equation

The list of applications that are utilized worldwide is shown below:

Real-world applications: There are several real-world scenarios where x2-11x+28=0 quadratic formula can be useful. It is used exclusively in astronomy to simulate possible orbital changes of celestial bodies under specific gravitational conditions.
Applications in science: In physics and engineering, this formula can be used to explain a variety of physical events involving curved trajectories or forms.

Mathеmatical Concеpts

The list of applications for the mathematical notion is shown below:

Formulas for quadratics: x2-11x+28=0 quadratic formula In this class of quadratic equations, the variable’s maximum power is two.
Radical extensions: The 3.2x radical extensions make the problem more complicated and call for a more specialized approach to solving it.

Importance of Quadratic Equations in Real Life

Quadratic equations like x² – 11x + 28 = 0 are not just academic exercises. They have real-life applications in various fields:

Physics: Quadratic equations are used to model projectile motion, calculate the trajectory of objects, and analyze forces in motion.
Economics: Quadratic functions are used to optimize profits, costs, and revenues.
Engineering: Structural engineers use quadratic equations to design stable structures and analyze forces on materials.

Conclusion

In this article, we’ve explored the different methods for solving the quadratic equation x² – 11x + 28 = 0: factoring, completing the square, and using the quadratic formula. Each method provides the same solutions, x = 4 and x = 7, but offers different approaches to solving the problem. By mastering these methods, students can confidently tackle quadratic equations and apply them in various academic and professional fields.