Solving a Quadratic Equation and Its Applications in Algebra
Quadratic equations are a fundamental part of algebra and have numerous real-world applications, from physics to engineering. In this article, we will delve into a specific quadratic equation and its solutions, as well as explore the significance of these solutions in algebraic contexts.
Understanding the Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, typically in the form (ax^2 bx c 0). The roots of a quadratic equation are the values of (x) that satisfy the equation. One such equation is (x^2 - 8x 16 0). Let's analyze and solve this equation step-by-step to uncover its roots and explore their significance.
Solving the Quadratic Equation
The given quadratic equation is:
(x^2 - 8x 16 0)To solve for (x), we can use the quadratic formula:
(x frac{-b pm sqrt{b^2 - 4ac}}{2a})Here, (a 1), (b -8), and (c 16). Plugging these values into the quadratic formula, we get:
(x frac{-(-8) pm sqrt{(-8)^2 - 4(1)(16)}}{2(1)})Evaluating the Discriminant
The discriminant ((Delta)) is given by (b^2 - 4ac). Calculating this:
(Delta (-8)^2 - 4(1)(16) 64 - 64 0)Since the discriminant is 0, the quadratic equation has a repeated root. Let's evaluate the equation for the repeated root:
(x frac{8 pm sqrt{0}}{2}) (x frac{8 pm 0}{2})Therefore, the repeated root is:
(x frac{8}{2} 4)Interpreting the Roots
The roots of the quadratic equation (x^2 - 8x 16 0) are both 4. This means that the equation can be factored as:
((x - 4)^2 0)Let's now apply this to two specific problems posed in the original prompt:
Problem 1: Finding (￡^5 - €^5)
Given that (￡ 4) and (€ 4), we need to find the value of (￡^5 - €^5).
(￡^5 - €^5 4^5 - 4^5 1024 - 1024 0)The value of (￡^5 - €^5) is 0.
Problem 2: Finding (￡ - €\)
Since both (￡) and (€\) are equal to 4, the difference (￡ - €\) is:
(￡ - € 4 - 4 0)The value of (￡ - €\) is 0.
Conclusion
By solving the quadratic equation (x^2 - 8x 16 0), we found that both roots are 4. This provided us with valuable insights into the nature of the roots and how to apply them to solve more complex problems. Understanding the properties of quadratic equations is crucial in algebra and various fields of mathematics and science.