**Solving the Quadratic Equation 4x^2 – 5x – 12 = 0**

Solving the quadratic equation **4x^2 – 5x – 12 = 0** is a fundamental skill in algebra that opens the door to understanding various mathematical concepts. In this article, we will explore different methods to find the solutions to this equation, delve into the significance of the discriminant, and discuss real-life applications of quadratic equations. So, let’s unravel the mysteries of quadratic equations together!

**Understanding the Quadratic Equation 4x^2 – 5x – 12 = 0**

A quadratic equation is a second-degree polynomial equation represented as ax^2 + bx + c = 0, where “x” is the variable, and “a,” “b,” and “c” are constants. In our equation, 4x^2 – 5x – 12 = 0, “a” is 4, “b” is -5, and “c” is -12.

**Method 1: Factoring**

Factoring is a primary technique to solve quadratic equations. The concept behind it is to rewrite the equation as a product of two binomials. For our equation:

**4x^2 – 5x – 12 = 0**

We can factor it as:

(2x + 3)(2x – 4) = 0

By setting each binomial to zero, we find the solutions:

2x + 3 = 0 or 2x – 4 = 0

Solving for “x” in each case, we obtain:

x = -3/2 or x = 2

**Method 2: Quadratic Formula**

The quadratic formula provides a direct and versatile way to find the solutions of any quadratic equation:

x = (-b ± √(b^2 – 4ac)) / 2a

For our equation, with a = 4, b = -5, and c = -12, the formula becomes:

x = (5 ± √(25 + 192)) / 8

Simplifying further:

x = (5 ± √217) / 8

The solutions are:

x = (5 + √217) / 8 or x = (5 – √217) / 8

**The Significance of the Discriminant**

In the quadratic formula, the term inside the square root, “b^2 – 4ac,” is known as the discriminant. The value of the discriminant provides crucial information about the nature of the solutions:

1. If the discriminant (b^2 – 4ac) is positive, two distinct real solutions exist.

2. If the discriminant is zero, there is one real solution (the graph touches the x-axis at a single point).

3. If the discriminant is negative, there are two complex solutions (no real solutions).

**Real-Life Applications of Quadratic Equations**

Quadratic equations have practical applications in various fields, including physics, engineering, economics, and computer graphics. Let’s explore some real-life scenarios where quadratic equations play a crucial role:

**Projectile Motion**

When an object is launched into the air and follows a parabolic trajectory, its path can be accurately described using a quadratic equation.

**Engineering and Design**

Quadratic equations are essential in designing structures like bridges, determining optimal shapes for various constructions, and predicting how materials will behave under different stress conditions.

**Finance and Economics**

In the financial world, quadratic equations find the production level that maximizes profits or minimizes costs, ensuring efficient resource allocation.

**Optics**

Quadratic equations are extensively used in optics, particularly in the design of lenses and mirrors, such as those found in cameras and telescopes.

**Conclusion**

In conclusion, solving the quadratic equation **4x^2 – 5x – 12 = 0** involves methods like factoring and the quadratic formula. Understanding the discriminant helps us interpret the nature of the solutions, which can be positive, singular, or complex. Quadratic equations play a significant role in various real-world applications across different fields, highlighting their relevance in modern science and technology. Mastering these algebraic techniques is a mathematical endeavor and a gateway to problem-solving in many practical domains.

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