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

In the realm of algebra, quadratic equations are among the most fundamental and widely encountered types of equations. They form the backbone of various mathematical concepts and find applications in numerous fields like physics, engineering, and finance. This detailed guide is dedicated to unraveling the method of solving a specific quadratic equation: 4x ^ 2 – 5x – 12 = 0, using the universally applicable quadratic formula. We will delve into each step with utmost clarity and precision, ensuring that even those new to quadratic equations can follow along and grasp the intricacies of this crucial mathematical process.

Understanding the Nature of Quadratic Equations

A quadratic equation is a second-degree polynomial equation typically presented in the standard form ax^2 + bx + c = 0. Here, ‘x’ represents the variable or the unknown, and ‘a’, ‘b’, and ‘c’ are coefficients with ‘a’ not equal to zero. The unique characteristic of quadratic equations is the presence of the x^2 term, which introduces the concept of curvature into the equation’s graphical representation. This curvature is what leads to the interesting property of quadratic equations having up to two real solutions.

In our specific case, the equation 4x^2 – 5x – 12 = 0, is a neatly structured quadratic equation where the coefficients are easily identifiable. Understanding these components is the first step in approaching any quadratic equation.

The Quadratic Formula: A Reliable Solver

Formula Breakdown

The quadratic formula is expressed as:

x = (-b ± √(b² – 4ac)) / 2a

This formula elegantly summarizes the solution to the quadratic equation ax^2 + bx + c = 0. The notation ‘±’ indicates that the equation usually has two solutions, corresponding to the two possible values obtained by either adding or subtracting the square root term.

The Discriminant: A Key Component

Central to the quadratic formula is the term ‘b² – 4ac’, known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has exactly one real root (or a repeated root). Lastly, if the discriminant is negative, the equation has no real roots but instead has two complex roots.

Solving 4x^2 – 5x – 12 = 0: A Step-by-Step Approach

Step 1: Identifying Coefficients

The first step in solving the equation 4x^2 – 5x – 12 = 0 using the quadratic formula is to identify the coefficients a, b, and c. In our equation, these are:

  • a = 4
  • b = -5
  • c = -12

Step 2: Applying the Quadratic Formula

Substitute the identified values into the quadratic formula:

x = (5 ± √(25 + 192)) / 8

Step 3: Simplifying the Equation

The next step involves simplifying under the square root:

x = (5 ± √217) / 8

This step is crucial as it brings us closer to finding the roots of the equation. The square root of 217 cannot be simplified further as 217 is not a perfect square. Thus, we proceed with the square root in its current form.

Step 4: Determining the Roots

The final step is to solve for the two possible values of x:

  • For x₁ (using the ‘+’ sign): x₁ = (5 + √217) / 8
  • For x₂ (using the ‘-‘ sign): x₂ = (5 – √217) / 8

These two values, x₁ and x₂, are the solutions to our equation 4x^2 – 5x – 12 = 0. They represent the points where the parabola, the graphical representation of a quadratic equation, crosses the x-axis.

Conclusion and Implications

By following the steps outlined above, we have successfully solved the quadratic equation 4x^2 – 5x – 12 = 0. The roots obtained, x₁ and x₂, are significant in that they provide valuable insights into the behavior of quadratic functions and their graphical representations. This method of using the quadratic formula is not just limited to this equation but can be universally applied to any quadratic equation, making it an indispensable tool in the arsenal of mathematicians and students alike.

Reflection and Application

The quadratic formula stands out for its elegance and universality in solving quadratic equations. Its ability to systematically handle any quadratic equation makes it a fundamental concept in algebra. Understanding and applying this formula is not only crucial for academic success in mathematics but also for practical problem-solving in science, engineering, and economics where modeling with quadratic equations is common.

Also Read: Brightspace Oswego: Redefining Education at SUNY Oswego


In conclusion, mastering the quadratic formula and its application, as demonstrated in solving the equation 4x ^ 2 – 5x – 12 = 0, opens doors to a deeper understanding of mathematical concepts and their real-world applications. It is a testament to the beauty and power of algebra in deciphering the complexities of the world around us.

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