Unraveling the Quadratic Mystery: Decoding 2×2-3x- 5 = 0

Introduction

The quadratic equation 2×2-3x- 5 = 0 is more than just a mathematical expression; it’s a cornerstone in algebra with wide-ranging applications. In this article, we’ll break down the intricacies of this quadratic equation, explore methods to find its roots, and discuss the practical significance of its solutions.

Understanding the Quadratic Equation: 2×2-3x- 5 = 0

Let’s delve into the quadratic equation 2×2-3x- 5 = 0, where the coefficients a, b, and c take on specific values. For this equation, a = 2, b = -3, and c = -5. Presented in the form ax² + bx + c = 0, it signifies a second-degree polynomial with the variable x.

Solving the Quadratic Equation

To discover the roots (solutions) of the quadratic equation, we turn to the quadratic formula:

x = (-b ± √(b² – 4ac)) / (2a).

Applying this formula to 2×2-3x- 5 = 0, we calculate the roots:

x = (3 ± √((-3)² – 4(2)(-5))) / (2(2)).

Simplifying further:

x = (3 ± √49) / 4.

This leads us to two potential solutions:

x = (3 + 7) / 4 = 2,

x = (3 – 7) / 4 = -1.

Graphical Representation

Visualizing the quadratic equation on a graph unveils a parabola. The points where the parabola intersects the x-axis signify the solutions. For 2×2-3x- 5 = 0, the roots are x = 2 and x = -1, representing the x-coordinates of these intersections.

Verification of Solutions

To validate the accuracy of our solutions, we substitute them back into the original equation. For x = 2 and x = -1:

2(2)² – 3(2) – 5 = 0,

2(-1)² – 3(-1) – 5 = 0.

Both evaluations result in 0, confirming that x = 2 and x = -1 are indeed solutions to the quadratic equation.

Conclusion

The quadratic equation 2×2-3x- 5 = 0 has been unraveled, and its roots, x = 2 and x = -1, have been determined through the quadratic formula. This equation, a cornerstone in algebra, holds relevance in various fields and lays the foundation for mathematical problem-solving. Mastering the methods to solve quadratic equations empowers individuals to tackle a myriad of mathematical challenges and real-world problems.