In the world of mathematics, equations are the building blocks of problem-solving. One such equation that often challenges students and mathematicians alike is the quadratic equation. In this article, we will delve into a specific quadratic equation: **2x^2 – 3x – 5 = 0**. We’ll break it down, explore its roots, and discuss its real-world applications. By the end of this journey, you’ll have a comprehensive understanding of this quadratic equation and its significance.

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### What is a Quadratic Equation?

Before we dive into the specifics of **2x^2 – 3x – 5 = 0**, let’s briefly recap what a quadratic equation is. A quadratic equation is a second-degree polynomial equation, meaning its highest power of the variable is 2. The general form of a quadratic equation is:

**ax^2 + bx + c = 0**

Here, ‘a,’ ‘b,’ and ‘c’ are coefficients, and ‘x’ represents the variable we’re solving for. In our case, ‘a’ is 2, ‘b’ is -3, and ‘c’ is -5.

## Solving 2x^2 – 3x – 5 = 0

### Step 1: Identify the Coefficients

In our equation, 2x^2 – 3x – 5 = 0, ‘a’ is 2, ‘b’ is -3, and ‘c’ is -5. These coefficients are crucial in solving the equation.

### Step 2: Use the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations:

**x = (-b ± √(b² – 4ac)) / (2a)**

### Step 3: Calculate the Discriminant

The discriminant, given by ‘b² – 4ac,’ plays a significant role in determining the nature of the roots.

### Step 4: Find the Roots

Depending on the value of the discriminant, we can classify the roots into three cases:

#### Case 1: Discriminant > 0

In this case, there are two distinct real roots.

#### Case 2: Discriminant = 0

Here, we have a single real root with multiplicity.

#### Case 3: Discriminant < 0

This case results in complex roots, which are conjugate pairs.

## Real-World Applications

Quadratic equations find their applications in various fields, including physics, engineering, economics, and even in our day-to-day lives. Some common scenarios where quadratic equations come into play include:

**Projectile Motion**: Calculating the trajectory of a thrown object.**Structural Engineering**: Analyzing the strength and stability of structures.**Finance**: Modeling investments and loan amortization.**Optics**: Designing curved mirrors and lenses.

## Conclusion

In this article, we’ve explored the quadratic equation 2x^2 – 3x – 5 = 0. We’ve learned how to solve it using the quadratic formula, discussed the significance of the discriminant, and touched upon real-world applications. Equipped with this knowledge, you can now tackle quadratic equations with confidence and apply them to solve real-life problems.