Quadratic equations open doors to a thrilling adventure in the mathematical world. In this blog, we aim to focus on solving one such equation, **4x ^ 2 – 5x – 12 = 0**. Specifically, quadratic equations significantly contribute to the algebraic concepts in various fields of science and mathematics. Without any further ado, let us dwell on the steps involved in solving this equation and its application to enlighten your minds with a better understanding.

**Quadratic Equation: A Brief Overview **

In short, quadratic equations are polynomial equations of the second degree. This implies they consist of variables raised to the power of two.

In its general form, it is represented as ax^2 + bx + c = 0.

Here, a, b, and c are coefficients, and x is the variable we are solving.

**Simplifying and Understanding the General Form**

In the specific equation we are looking to solve, **4x ^ 2 – 5x – 12 = 0**, the coefficients are as follows:

- A = 4
- B+ -5
- C= -12

When we rearrange this equation, we can identify it as the general form of a quadratic equation.

**Characteristics of Quadratic Equations**

Quadratic equations display certain characteristics that significantly contribute to their study and solution. These characteristics include:

- A quadratic equation has a parabolic shape
- It has a vertex representing the minimum/maximum point
- It has symmetrical properties

Understanding these characteristics opens up the doors to provide us with a better insight into their behavior and applications in the real world.

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

Different formulas and methods exist to solve the equation **4x ^ 2 – 5x – 12 = 0**. We are going to explore two different methods:

- Factoring Method
- Quadratic Formula

Let us now explore each of them to gain a better understanding.

**Factoring Method**

You can use the factoring method only when the equation is factorable. It works by breaking down the quadratic equation into its factors, which can be solved individually.

To solve the equation **4x ^ 2 – 5x – 12 = 0** using the factoring method, you must find two binomials to multiply to give us the quadratic equation.

The factored form would be (2x + 3) (2x – 4) = 0.

Equating each factor to zero will obtain two equations: 2x + 3 = 0 and 2x – 4 = 0.

Once you solve both of these equations, you will find the values of x. Hence, the solution to the equation will be x= -3/2 and x=2.

**Quadratic Formula**

An alternative method to solve quadratic equations is the quadratic formula. It is one of the most renowned formulas and a powerful tool that guarantees a solution, even when the equation is not easily factorable.

For the equation **4x ^ 2 – 5x – 12 = 0**, we can use the quadratic formula:

**x = (-b ± √(b^2 – 4ac)) / 2a**

In the formula above,

- a = 4
- b = -5
- c = -12

After plugging this into the formula, we can easily calculate and find the solutions for x.

After calculating, we find that x = -3/2 and x = 2.

**Real World Applications**

Quadratic equations have many applications in several fields, including economics, engineering, and physics. Some real-world scenarios where quadratic equations play an important role include Engineering, Physics, and Projectile Motion.

**Consider Reading:** Value of x**x**x is Equal to 2: A Comprehensive Guide

**Conclusion**

We have learned the quadratic equation **4x ^ 2 – 5x – 12 = 0** and explored two methods to solve it: the factoring method and the quadratic formula. Using these methods, we were able to find the solution to x. It is important to understand the quadratic equations to solve a wide range of mathematical problems.

**Frequently Asked Questions (FAQs)**

**What is a Quadratic Equation?**

A quadratic equation is defined as a polynomial equation of a second degree. It implies that the equation comprises at least one term that is squared.

**Who is known as the father of the quadratic equation?**

Muhammed ibn Musa al-Khwarizmi.