4x ^ 2 – 5x – 12 = 0 : Quadratic Equation Solution

4x ^ 2 - 5x - 12 = 0

Here we are going to explore and solve the amazing quadratic problem 4x ^ 2 – 5x – 12 = 0. These days, Quadratic Equation is an extremely important part in the mathematics syllabus of every institution, be it schools or universities.

Although many may think that Quadratic does not have any practical uses, it is actually nothing but a misconception because everything has its own important uses is many other fields of study that are more oriented in the research and development fields.

Quadratic Equation

In the world of algebra, the quadratic equation is one of the most important aspects of it. This category in algebra is considered to be one of the fundamental concepts and without leaning this, one may miss out on a hugely important part of learning about an important part of mathematics. In our article today, we will be taking a better look at what quadratic equation is all about and below we have provide an comprehensive solution of 4x2 – 5x – 12 = 0.

Concept of Quadratic Equation

A quadratic equation can be defined as a polynomial equation that is of second degree, and this basically means that the equation has at least one term that is squared. The general form of the quadratic equation is something like this ax2 + bx + c = 0

where with a, b and c are constants and x represents the variable. We have an equation to solve through this article and then keep it aside. The equation that we are required to solved and that is.

Q. Equation :- 4x ^ 2 – 5x – 12 = 0

This is an equation that one can solve easily with the help of quadratic equation formulas. There are two ways this equation can be solved, one is the direct method and the other is the Sridhar Acharya method However, since this equation is pretty best to just go for the direct approach.

Solution 4x ^ 2 – 5x – 12 = 0:

If you are planning to go for the direct approach, there are two values of x that you will get. You will also be getting two values for x. Here is a step by step guide on how you can solve this equation easily. Keep reading to find out:

First step would be to write the equation newly. Just copy down the equation from your question paper or book into your notebook.

  • Then, write down your equation in this form:


  • Once this is done, break the middle part of the equation and write it like this:


  • Next, write your equation like this after taking common from both parts:


But you will see that the values you get from equating this far are not desirable and hence, the above solution is not valid.

That being said, there is another way of going this sum, and that is by utilizing the Sridhar Acharya method. In that method, you will need to use a formula, and that formula is

Here, we will need to substitute x, a and c with the necessary values from the equation and then place it here in these equation. Eventually, as you keep calculating, you will find the necessary values.

  • x = (-b ± √(b2 – 4ac)) / 2a
  • Put a= 4, b=5 and c= 12
  • x=(-5 ± √(52 – 4x4x12))/2×4
  • After solving the above equation, we get x= √217 + 5/8 or x= – √217 + 5/8
  • And, after calculating the root value we get
  • Axis of Symmetry (dashed)  {x}={ 0.62} 
  • Vertex at  {x ,y} = { 0.62,-13.56} 
  •  x -Intercepts (Roots) :
  • Root 1 at  {x ,y} = {-1.22, 0.00} 
  • Root 2 at  {x ,y} = { 2.47, 0.00}

Therefore, the Quadartic equation 4x2 – 5x – 12 = 0 gives the values for x, which is x =5+√2178 , x =5−√2178 or x =2.4663 x =−1.21636 in decimal.

Graph For 4x2 – 5x – 12 = 0

Uses Of Quadratic Equation

Quadratic equations may look like something that does not have much use in today’s world but in reality this equation, and the formulas of it”, are very important for several other things. Below you will find list of places where quadratic equations are extremely useful:

  • Physics is one of the fields where quadratic equations are utilized for performing complex calculations. Quadratic equations are utilized for calculating equations of projectile motion and other important topics of physics.
  • Engineering uses quadratic equations very often for solving equations. These equations can be regarding signal processing, structural analysis or for electrical circuits as well.
  • It may be strange to hear, but the study of Finances uses quadratic equations, especially in the areas of modeling complex financial systems or figuring out tax investment returns

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