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4x ^ 2 – 5x – 12 = 0: A Quadratic Equation And Its Solutions

An Explanation For Quadratic Equations?

A polynomial equation of the second degree with at least one squared term (x^2) is present then it is a quadratic equation. ax^2 + bx + c = 0 is a general form of a quadratic equation, in its general form a,b and c are constant and x is variable. Similarly we are going to apply this method on your equation that is 4x ^ 2 – 5x – 12 = 0.

Equation in Quadratic Standard Form

It is necessary to know a quadratic equation’s standard form in order to determine its leading coefficient. A quadratic equation is written as ax^2 + bx + c = 0 in standard form. 

Getting into 4x ^ 2 – 5x – 12 = 0:

 ax^2 + bx + c = 0 is a mathematical expression for a quadratic equation and by following these expressions we can justify our quadratic equation in which a, b and c will remain constant and x will represent a variable.

How to Solve A Quadratic Equation

The equation can be solved by following several methods and below are three of them which is mainly used for solving:

Solving By Factoring Method:

The quadratic equation can be written using the factoring method as the sum of two binomials. When the problem can be easily factored, this strategy works very well. If we put the roots of every binomial at 0 and figuring out what x can be, we can find the roots.

Solving By Quadratic Formula:

It is known as the universal method for getting the resolved answer of any quadratic equation. We can substitute a, b and c in formula x = (-b ± √(b^2 – 4ac)) / 2a from our equation which is 4x ^ 2 – 5x – 12 = 0.

Solving By Completing the Square

The other very useful method for solving the quadratic equation is this. The process of this method is transforming the particular equation in a trinomial square which simplifies the roots to solve it easily.

Root Patterns with Discriminant Methods


In relation with any quadratic equation, to understand the nature of roots this discriminant works as an important category. b^2 – 4ac is the given expression and we can have the roots as real and diverse, real and similar or others which depend on the discriminant value.
Below are mentioned each steps:

Getting Real and Diverse Roots

We get two different diverse roots from the quadratic equation when the value of discriminant is more than 0. In a graphical presentation, it represents the points where the equation intersects the x-axis.

Getting Real and Similar Roots

Following the same graphical method, the equation meets at a common point on x-axis. It happens when the discriminant value is equal to 0.

Uses of Quadratic Equations

The uses of Quadratic equation in various field and disciplines are very important and some of them are:

Uses in Physics

It helps to resolve any issues related to projectile motions which basically means moving an object from one place to another or a displacement of any object.

Uses in Engineering and Design

It helps in terms of analyzing any crucial determination such as load distribution in structure analysis, electrical circuits and processing the signal. So it is very useful for engineers in their particular fields.

Uses in Economics and Finance:

It helps in utilizing the way for getting the approx return on investment and modify the financial systems in case of complex economical structure.


4x^2 – 5x – 12 = 0 is a quadratic equation which gives a good example of a polynomial equation of the second degree. In this blog we have covered each important step which can make the equation easy for getting the solution. We have also mentioned the practical aspects of the quadratic equation in our day today life. Since rather than this there are more features and applications of quadratic equations which can be found by studying more and more.

Frequently Asked Questions:

Q. In a quadratic equation, what is its root?

Ans: The determined value of the x is known as the roots of the quadratic equation and it can be found by following the shown methods in the blog such as quadratic formula, factoring method and others.

Q. Do quadratic equations contain complex roots?

Ans: Absolutely, If the determined value of the discriminant is less than 0 and its in negative term then it can contain the complex roots.

Q. Do quadratic equation applications work in day to day life?

Ans: It has multiple crucial roles in our day to day life and also plays an important role in various fields.Mainly in terms of physics, engineering and design, economics and finance and others.

Q. How can we understand Quadratic equations in a more precise manner?

Ans: There are several sources which provide valuable guidance on solving any quadratic equations and some common sources are mathematics books, mathematics youtube channels or any teaching website.

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