A warm welcome to all readers! In this blog we are going to discuss everything related to the Quadratic Equation and the solution for 4x ^ 2 – 5x – 12 = 0. In terms of algebra, the quadratic equation and its methods play an important role in mathematics. Let’s dive in to know more about the step by step solution with examples.
What is a Quadratic Equation?
Quadratic equations are math questions with the second degree. This means they have numbers times a letter like x^2 ( square of x ). Quadratic Equations is a polynomial equation that is powered by second degree, this means it contains variables raised to the power of two. They look like this: ax^2 + bx + c = 0. Here we will work on getting the answer for x, whereas a, b, and c are just numbers.
The General Form:
If we review the shape of the quadratic problems then by changing the order of the equation, In this case, we have an equation 4x ^ 2 – 5x – 12 = 0, whereas a is 4, b is -5, and c is -12.
Quadratic Equations Features:
Quadratic problems are special because they have certain things we need to know. The shape originated from the top and bottom point that is a vertex. We can simply understand it as two corner of the playground
Solution for 4x ^ 2 – 5x – 12 = 0 :
There are two very known methods which can be used for solving the equation 4x^2 – 5x – 12 = 0, the first one is the quadratic formula and the second one is the factoring method.
1. Quadratic Formula:
For solving any quadratic problems this quadratic formula provides very effective methods. To find the answer it works like a very strong tool, even if the solution is very tough to get.
For the problem 4x^2 – 5x – 12 = 0, we can use the quadratic formula:
x = (-b ± √(b^2 – 4ac)) / 2a
We get x = -3/2 and x = 2 by putting the numbers a = 4, b = -5, and c = -12 into this formula, we can work out the answers for x.
For 4x^2 – 5x – 12 = 0:
x = (-(-5) ± √((-5)^2 – 4 * 4 * (-12))) / (2 * 4)
Hence, the solution we get after calculating the above equation is x = -3/2 and x = 2.
2. Factoring Method:
The process of the Factoring method is dividing the quadratic equation into multiple factors and then solving each factor individually. Although the requirement is that the equation should be factorable.
(2x + 3)(2x – 4) = 0 will be the generated factor for the quadratic equation 4x^2 – 5x – 12 = 0, we needed the two binomials to give the required equation by multiplying together.
Getting Factors of 4x^2 – 5x – 12 = 0:
(2x + 3)(2x – 4) = 0
Equating of each factor with 0, we get:
2x + 3 = 0 –> x = -3/2
2x – 4 = 0 –> x = 2
Hence, the solutions we get from following the factoring method are x = -3/2 and x = 2.
Once we work on both factors individually by equating with 0 then we get two equations that are 2x + 3 = 0 and 2x – 4 = 0. Next, when we solve these two equations then we get the final solution of x that is x = -3/2 and x = 2.
Uses of Quadratic Equations:
There are multiple uses of quadratic equations in each field and in our day to day life. The various fields can be physics, engineering, economics or others. Below are some of the major examples for it:
Defining Thrown Movement:
To understand the concept of throwing weather from up to down or one place to another place Quadratic equations plays an important role in that. The displacement position of an object such as baseball, fired bullet or launched rocket can be defined by using the quadratic equation. Taking some of the physical terms like initial velocity, angle of projection or any mass and acceleration that is force working on the object and others can be obtained by using the Quadratic equation.
Study of Engineering and Physics
In the world of engineering and physics there are many crucial terms such as resolving the issues of maintenance, movement of objects and getting the equilibrium which can be only obtained by using the quadratic equation. Particularly in engineering when we talk about designings and structural stability then the equation provides the important formulas which can be used to overcome any issues.
Graphical Representation of 4x ^ 2 – 5x – 12 = 0
It’s another way to represent this quadratic equation on a graph. Below mentioned is a pictographic way to show:
- Vertex Form
The vertex form of any quadratic equation can be written as f(x) = a(x-h)^2 + k. Here (h,k) shows the vertex of parabola. Then, converting this equation into vertex format allows users to determine the vertex and other significant properties of graph.
- Axis of Symmetry
This is a vertical line that passes through the vertex of parabola. It enables to divide the parabola into two symmetrical halves.
- Mapping the graph
Graphical representation provides an intrigated visualization of the quadratic equation.
In this blog we have reviewed the solutions for the quadratic equation 4x^2 – 5x – 12 = 0 by utilizing a couple of methods that are quadratic formula and factoring methods and determining the actual required values. Also we have understood the practical aspects of the quadratic equation in different fields and in day to day life.