Quadratic equations are a fundamental part of algebra, and they appear in various aspects of mathematics and science. In this comprehensive article, we will delve deeply into the quadratic equation 4x ^ 2 – 5x – 12 = 0, exploring its solutions and providing valuable insights on how to tackle similar equations.

## Understanding Quadratic Equations 4x ^ 2 – 5x – 12 = 0

Quadratic equations are polynomial equations of the second degree, and they have a standardized form: ax^2 + bx + c = 0. Here, ‘a,’ ‘b,’ and ‘c’ are constants, and ‘x’ represents the variable. The equation we’ll be focusing on, 4x ^ 2 – 5x – 12 = 0, is indeed a quadratic equation.

Quadratic equations are ubiquitous in mathematics and science because they model various real-world phenomena, from the motion of projectiles to the growth of populations. Hence, having a profound understanding of how to solve them is crucial for mastering these fields.

## The Importance of Quadratic Equations 4x ^ 2 – 5x – 12 = 0

Quadratic equations find applications in diverse disciplines, making them indispensable in the world of mathematics and beyond. Let’s explore why they are so important:

**Physics**: In physics, quadratic equations are used to describe the motion of objects under the influence of gravity. For instance, when calculating the trajectory of a launched projectile, we often encounter quadratic equations.**Engineering**: Engineers employ quadratic equations when designing structures, analyzing electrical circuits, and predicting mechanical behaviors. They provide essential tools for optimizing designs and ensuring the safety and functionality of various systems.**Economics**: In economics, quadratic equations are used to model revenue, cost, and profit functions. Businesses use them to determine the price points that maximize profits or minimize costs.**Biology**: Biologists use quadratic equations to study population dynamics, the growth of organisms, and the spread of diseases. These equations help researchers make predictions and formulate strategies for managing biological systems.

## Solving 4x ^ 2 – 5x – 12 = 0

### Factoring

One common method to solve quadratic equations is factoring, a technique that involves rewriting the equation in a way that makes it easier to find the values of ‘x’ that satisfy it. To factor the quadratic equation 4x ^ 2 – 5x – 12 = 0, we need to find two numbers whose product equals the product of the leading coefficient ‘a’ and the constant term ‘c’ (in this case, 4 * -12 = -48) and whose sum equals the coefficient of ‘x’ (‘b,’ which is -5). These numbers are -8 and 6.

So, we can rewrite the equation as follows: (4x^2 – 8x) + (3x – 12) = 0.

### Grouping and Factoring

To proceed, we group the terms with common factors and factor them separately: 4x(x – 2) + 3(x – 4) = 0.

Now, we can see that both terms share a common factor, which is (x – 2): (x – 2)(4x + 3) = 0.

## Solving for ‘x’

With the equation factored, we can now find the solutions by setting each factor equal to zero:

- x – 2 = 0
- 4x + 3 = 0.

Solving these equations separately, we find the solutions:

- x = 2
- 4x = -3 x = -3/4.

## Conclusion

In conclusion, understanding how to solve quadratic equations, such as 4x ^ 2 – 5x – 12 = 0, is a valuable skill with wide-ranging applications in mathematics, science, and real-life scenarios. We have explored the factoring method as a powerful tool to find the solutions to such equations, but other methods like the quadratic formula and completing the square can also be employed.