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In the realm of mathematics, x2-11x+28=0 quadratic equations occupy a significant and intriguing position. They manifest themselves in various domains, from the realms of physics and engineering to everyday problem-solving scenarios. Among these quadratic equations, x2-11x+28=0 is a noteworthy one. This article aims to delve deep into the intricacies of this equation, 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! Continue reading to demystify its enigma.
Breaking Down the Equation x2-11x+28=0
Before we embark on our journey to solve the enigmatic x2-11x+28=0, it’s essential to deconstruct this equation into its fundamental components.
The Quadratic Equation x2-11x+28=0
Our equation adheres to the standard format of a quadratic equation, which can be expressed as Ax² + Bx + C = 0. In the case at hand, the values are as follows: A = 1, B = -11, and C = 28.
What are Quadratic Equations?
Quadratic equations, in their essence, are polynomial equations of the second degree. Graphically, they often manifest as parabolic curves. As for our equation, x2-11x+28=0, it represents a specific parabola in its graphical representation.
Finding the Roots
The primary objective when dealing with a quadratic equation like x2-11x+28=0 is to determine its roots. These roots signify the values of ‘x’ that render the equation equal to zero.
Factoring the Equation
One of the tried and tested methods employed for solving quadratic equations is the process of factoring. In the subsequent sections, we will elucidate this approach in detail, leading us to the discovery of the roots of
x-11x+28=0.
Factoring the Quadratic Equation
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 the discovery of 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, we can now embark on the journey to uncover its roots. This is achieved by setting each factor equal to zero:
- x – 7 = 0
- 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:
- x = 7
- x = 4
Conclusion
In the culmination of our exploration, we have effectively unraveled the mysteries of the quadratic equation x2-11x+28=0 through the process of factoring. The roots of this equation emerge as x = 7 and x = 4. This achievement underscores the fundamental nature of quadratic equations in mathematics and their diverse applications across various domains.
