Solving the Equation 2×2-3x- 5 = 0. When we embark on the journey of understanding algebraic equations, quadratic equations take center stage due to their significance and complexity. One such intriguing quadratic equation that often challenges students is the equation 2×2-3x- 5 = 0. This article will serve as your compass, guiding you through the intricate process of solving this quadratic puzzle step by step. Let’s dive into the depths of mathematics and unlock the methods to conquer this equation with confidence.
Unraveling The Quadratic Equations 2×2-3x- 5 = 0
Before we delve into the intricacies of solving the given quadratic equation, it’s crucial to establish a solid foundation regarding what exactly a quadratic equation entails. In its general form, a quadratic equation is a polynomial equation of the second degree, involving a single variable x. The standard representation takes the form ax2 + bx + c = 0, where a, b, and c are constants, with the stipulation that a ≠ 0. In our specific equation, the coefficients are a = 2, b = -3, and c = -5.
Exploring Various Approaches to Solve Quadratic Equations
1. Factoring: Unveiling the Building Blocks
The process of factoring involves breaking down a complex expression into simpler components, akin to deconstructing a puzzle. However, in the case of the equation 2×2-3x- 5 = 0, the coefficients don’t readily align to yield integer solutions, necessitating exploration of alternative methods.
2. Unveiling the Quadratic Formula: A Mathematical Eureka Moment
The quadratic formula emerges as a potent weapon in our mathematical arsenal, capable of providing solutions to any quadratic equation. For an equation of the form ax^2 + bx + c = 0, the quadratic formula unfurls as follows: x = (-b ± √(b^2 – 4ac)) / 2a
Implementing this formula with the coefficients from our equation leads us to: x = (3 ± √((-3)^2 – 4(2)(-5))) / 2(2)
Calculating the discriminant, represented as D: D = (-3)^2 – 4(2)(-5) = 9 + 40 = 49
This journey culminates in two potential solutions for x.
Completing the square presents itself as an alternate route to conquer quadratic equations. By transforming the equation into a perfect square trinomial and skillfully isolating x, this approach unveils its elegance.
4. The Art of Graphical Analysis
Leveraging the power of graphical representation, we can gain invaluable insights into the roots of quadratic equations. The points at which the graph intersects the x-axis unravel the hidden solutions, painting a visual masterpiece of mathematical exploration.
Deciphering 2×2-3x- 5 = 0: A Step-by-Step Odyssey
Embarking on a mathematical odyssey, let’s dissect the enigma of the quadratic equation 2×2-3x- 5 = 0 by employing the aforementioned methods:
Step 1: Embracing the Quadratic Formula
As previously revealed, the quadratic formula stands as follows: x = (-b ± √(b^2 – 4ac)) / 2a
Substituting the coefficients from our equation yields: x = (3 ± √((-3)^2 – 4(2)(-5))) / 2(2)
Evaluating the discriminant, denoted as D: D = (-3)^2 – 4(2)(-5) = 9 + 40 = 49
Consequently, the solutions for x manifest as: x = (3 + √49) / 4 or x = (3 – √49) / 4
Further simplification ensues: x = (3 + 7) / 4 or x = (3 – 7) / 4 x = 10 / 4 or x = -4 / 4 x = 2.5 or x = -1
Step 2: The Graphical Voyage
To validate our solutions, let’s set sail on a graphical expedition and chart the equation 2×2 – 3x – 5 = 0. The intersection points of the curve and the x-axis come to life, closely resembling x = 2.5 and x = -1, corroborating our previous calculations.
A Resolute Conclusion
Navigating the labyrinth of quadratic equations, exemplified by 2×2-3x- 5 = 0, entails a diverse array of methodologies, from the robust quadratic formula to the enlightening graphical analysis. Each avenue unveils a distinct perspective on the solutions. In this expedition, we have unveiled that the solutions approximately take the forms of x = 2.5 and x = -1.