Solving the Quadratic Equation: 2x^2 – 3x – 5 = 0

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In the vast world of mathematics, quadratic equations hold a special place. These equations, typically represented in the format (ax^2 + bx + c = 0), are a frequent appearance in various disciplines, from physics to finance. Today, we’re diving deep into one such equation: (2x^2 – 3x – 5 = 0). Let’s unravel its solutions and understand its significance.

1. Understanding the Quadratic Equation

Before we dive into the solution, it’s essential to get a grip on what a quadratic equation is. In simple terms, it’s a polynomial equation of degree 2. The general form is:

[ ax^2 + bx + c = 0 ]

Where:

  • (a), (b), and (c) are constants
  • (a ≠ 0); if (a = 0), then it would be a linear equation

The equation we’re exploring today is (2x^2 – 3x – 5 = 0), where:

  • (a = 2)
  • (b = -3)
  • (c = -5)

2. Solving the Quadratic Equation

For quadratic equations, one popular solution method is the Quadratic Formula:

[ x = \frac{-b ± \sqrt{b^2-4ac}}{2a} ]

Plugging in the values from our equation:

[ x = \frac{3 ± \sqrt{9 + 40}}{4} ]

[ x = \frac{3 ± \sqrt{49}}{4} ]

[ x = \frac{3 ± 7}{4} ]

This gives us two solutions:
[ x_1 = \frac{10}{4} = 2.5 ]
[ x_2 = \frac{-4}{4} = -1 ]

3. Graphical Representation

If you were to graph the equation (2x^2 – 3x – 5), it would be a parabola opening upwards (since (a > 0)). The solutions (x = 2.5) and (x = -1) are the points where this parabola intersects the x-axis.

4. Significance in Real Life

Quadratic equations aren’t just theoretical entities; they have practical applications too. For instance, they can describe the trajectory of a thrown object, the time it takes for an investment to double, or even how sales might change relative to a price increase or decrease. Understanding how to solve them and what the solutions represent can be crucial in decision-making processes across various sectors.

Conclusion

Mathematics offers tools to understand and interpret the world around us. The quadratic equation (2x^2 – 3x – 5 = 0) might seem like a set of random numbers and symbols to some, but as we’ve seen, it holds particular solutions and can represent real-world phenomena. By mastering equations like this, we can better navigate the complexities of both abstract and tangible challenges.

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