Solve Quadratic equations: 4x^2-5x-12=0

4x^2-5x-12=0

We shall explore the interesting realm of quadratic equations in this article. We will specifically concentrate on finding the solution to 4x^2 – 5x – 12 = 0. Quadratic equations are a fundamental algebra component important in many branches of science and mathematics. To improve our comprehension, we will review the procedures for solving this equation, discuss its uses, and present examples throughout this piece.

4x ^ 2 - 5x - 12 = 0
4x ^ 2 – 5x – 12 = 0

This quadratic equation is an example: 4x^2 – 5x – 12 = 0. This equation is mathematical, and its solution calls for a formula. This is an example of an equation-based mathematical question where numbers and variables are provided. All we need to do is demonstrate that the left and right sides of this equation are equal (L.H.S. = R.H.S.).

One of the forms of a quadratic equation is 4x^2 – 5x<12 = 0. Usually, the simplification approach is used to tackle problems involving quadratic equations of this kind. Equations are broken down using this strategy until the equation equalizes. Although mathematical equations appear simple, they are rather challenging. To answer any equation, you need to grasp some fundamental math concepts and formulae.

What is a Quadratic Equation?

Second-degree polynomial equations, or those involving variables raised to the power of two, are known as quadratic equations. A quadratic equation can be written in the general form as ax^2 + bx + c = 0, where x is the variable we are solving for and a, b, and c are the coefficients.

A mathematical phrase is a quadratic equation. The purpose of a quadratic equation is to demonstrate a solution through problem-solving. The word comes from the Latin quadratus. The quadratic equation 4x^2 – 5x – 12 = 0 is an excellent illustration. a second-degree polynomial equation with a square in at least one of the variables.  The formula for quadratic equations is ax^2 + bx + c = 0.

For instance:

  • In the given equation, ax^2 + bx + c = 0
  • a, b, c are called a known coefficient where a is not equal to 0
  • x is represented as a variable.

Understanding the General Form

The coefficients in our particular equation, 4x^2 – 5x – 12 = 0, are a = 4, b = -5, and c = -12. We can observe that the equation follows the general form of a quadratic equation by rearranging it.

Characteristics of Quadratic Equations

Certain features of quadratic equations are essential to understanding and solving them. These consist of having symmetrical characteristics, a vertex that indicates the minimum or maximum point, and a parabolic form. Gaining knowledge of these attributes enables us to comprehend their behavior and useful applications.

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

There are various techniques we can use to solve the equation 4x^2 – 5x – 12 = 0, including the quadratic formula and the factoring approach. Let’s investigate each of these methods.

The process of factoring Method

The factoring method involves breaking down the quadratic equation into its factors, which can be solved individually. However, this method works only when the equation is factorable.

To solve 4x^2 – 5x – 12 = 0 using the factoring method, we need to find two binomials that multiply together to give us the quadratic equation. In this case, the factored form would be (2x + 3)(2x – 4) = 0.

By equating each factor to zero, we obtain two equations: 2x + 3 = 0 and 2x – 4 = 0. Solving these equations gives us the values of x. Therefore, x = -3/2 and x = 2 are the solutions to the equation.

Formula for Quadratic Formula

An alternative approach to solving quadratic problems is offered by the quadratic formula. Even in cases where factoring an equation is difficult, this effective technique ensures that the answers will be found.

The quadratic formula can be applied to the problem 4x^2 – 5x – 12 = 0.

x equals (-b± √(b^2 – 4ac)) / 2a.

We can find the solutions for x by entering the variables a = 4, b = -5, and c = -12 into the formula. Upon completing the required computations, we determine that x = -3/2 and x = 2.

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Examples of Solving 4x^2 – 5x – 12 = 0

To find the coefficients, use the standard form of a quadratic equation:

Conclusion

Finally, we have studied many approaches to solving the quadratic problem 4x^2 – 5x – 12 = 0. We can determine the values of x that fulfill the equation by applying the quadratic formula or the factoring approach. In addition, we now know that quadratic equations have practical uses in disciplines like physics, engineering, and projectile motion. Comprehending quadratic equations is crucial for resolving an extensive array of mathematical issues and offers a significant understanding of the actions of diverse occurrences.

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