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Reliable Solution for Equation: x2-11x+28=0, Quadratic equations form the cornerstone of algebra and pave the way for understanding more complex mathematical concepts. The equation x2-11x+28=0 is a classic example, demonstrating not only the beauty of algebra but also the systematic approach required to solve quadratic equations. This article aims to guide you through the process of solving this equation, ensuring a deep understanding of both the theory and application of algebraic principles.

**Understanding Quadratic Equations**

At the heart of algebra, quadratic equations are polynomial equations of the second degree. The general form is ax^2 + bx + c = 0, where a, b, and c are constants. These equations are pivotal in various fields, including engineering, physics, and economics, due to their ability to model numerous natural and man-made phenomena.

**The Quadratic Formula: A Reliable Solution Method**

The quadratic formula, derived from the process of completing the square, provides a direct method for solving any quadratic equation. For the equation ax^2 + bx + c = 0, the roots can be found using the formula:

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

This formula is not just a mathematical artifact; it is a testament to human ingenuity, offering a glimpse into the symmetry and structure underlying algebraic equations.

**Step-by-Step Solution of x2-11x+28=0**

To solve the equation x^2 – 11x + 28 = 0 using the quadratic formula, we first identify the coefficients a = 1, b = -11, and c = 28. Plugging these values into the formula yields:

$x=\frac{-(-11)\pm \sqrt{(-11{)}^{2}-4(1)(28)}}{2(1)}$

$x=\frac{11\pm \sqrt{121-112}}{2}$

$x=\frac{11\pm \sqrt{9}}{2}$

$x=\frac{11\pm 3}{2}$

This leads to two possible solutions for x:

${x}_{1}=\frac{11+3}{2}=7$

${x}_{2}=\frac{11-3}{2}=4$

Therefore, the solutions to the equationx2-11x+28=0 are x = 7 and x = 4.

**Applications of Quadratic Equations in Real Life**

The practical applications of quadratic equations are vast and varied. From calculating areas to determining the trajectory of objects under the force of gravity, these equations are indispensable tools. For instance, in finance, they can model profit maximization problems, while in physics, they describe the motion of particles in potential fields.

**The Importance of Algebraic Skills**

Developing a strong foundation in algebra is not merely an academic exercise; it is an investment in critical thinking and problem-solving skills. The ability to manipulate and solve equations like x2-11x+28=0 equips students with the tools to tackle a wide range of challenges, both in their academic pursuits and in their everyday lives.

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**Conclusion**

In conclusion, the equation x2-11x+28=0 serves as a powerful example of the elegance and utility of algebra. By understanding and applying the quadratic formula, students can solve this and similar equations, unlocking the doors to a deeper comprehension of the mathematical world. The journey through algebra is not just about finding the right answers; it’s about cultivating a mindset that appreciates the intricate patterns and structures that govern our universe.

**Further Reading and Resources**

For those eager to deepen their understanding of algebra and its applications, numerous resources are available. Textbooks, online courses, and interactive tutorials offer pathways to enhance mathematical knowledge and skills. Additionally, engaging with mathematical problems and solutions in forums and study groups can provide valuable insights and foster a love for the subject.