## 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 axÂ² + 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 Â± âˆš(bÂ² – 4ac)) / (2a).

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

x = (3 Â± âˆš((-3)Â² – 4(2)(-5))) / (2(2)).

Simplifying further:

x = (3 Â± âˆš49) / 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)Â² – 3(2) – 5 = 0,

2(-1)Â² – 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.