People often consider mathematics a language unto itself. And as with any language, we must understand specific rules, patterns, and equations to communicate effectively. One such equation is the quadratic equation, specifically, 4x^2 – 5x – 12 = 0. This article will explore the nuances of this equation and guide you through the steps to solve it.
What is a Quadratic Equation?
Before we delve into solving 4x^2 – 5x – 12 = 0, it’s essential to understand what a quadratic equation is. A quadratic equation is a second-order polynomial equation in a single variable, x, with a degree of two. The general form is ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
The quadratic equation is known for its distinctive ‘U’ shaped graph, a parabola. The solutions of the quadratic equation are the x-coordinates where the parabola crosses the x-axis.
Breakdown of the Quadratic Equation: 4x^2 – 5x – 12 = 0
The equation 4x^2 – 5x – 12 = 0 perfectly fits a quadratic equation’s standard form. Here, a=4, b=-5, and c=-12. These values give the equation its unique shape and solutions. The solutions of the equation are the roots of the equation, the points where the graph intersects with the x-axis.
Solving the Quadratic Equation: A Step-by-step Guide
Several methods for solving quadratic equations exist, such as factoring, completing the square, and using the quadratic formula. For our equation 4x^2 – 5x – 12 = 0, we will use the quadratic formula, which is the most general approach.
The quadratic formula is:
x = [ -b ± sqrt(b^2 – 4ac) ] / (2a)
Step 1: Identify a, b, and c
As mentioned, in our equation 4x^2 – 5x – 12 = 0, a=4, b=-5, and c=-12.
Step 2: Plug the values into the quadratic formula
We insert our values into the quadratic formula. So we have:
x = [ 5 ± sqrt((-5)^2 – 44(-12)) ] / (2*4)
Step 3: Simplify
We simplify within the square root first:
x = [ 5 ± sqrt(25 – (-192)) ] / 8
x = [ 5 ± sqrt(217) ] / 8
Then, we find two possible solutions:
x1 = (5 + sqrt(217)) / 8
x2 = (5 – sqrt(217)) / 8
These are the roots of our equation 4x^2 – 5x – 12 = 0.
Understanding the Solutions
A quadratic equation can have two, one, or no real solutions, depending on the value of the discriminant (b^2 – 4ac). If the discriminant is positive, as in our case, we get two distinct real solutions. If it’s zero, we get one real solution; if it’s negative, we get no real answers – instead, we get two complex solutions.
The solutions represent the x-values where the parabola crosses the x-axis. These solutions could represent times, distances, or quantities in practical applications, depending on the problem.
The Role of Quadratic Equations in Daily Life
To the uninitiated, 4x^2 – 5x – 12 = 0 may seem like a random assortment of numbers and symbols. However, quadratic equations are vital in various fields, from physics to economics. They can model the trajectory of a ball, determine the optimum price for a product, or predict population trends, to name a few examples.
Conclusion
In this exploration of the quadratic equation 4x^2 – 5x – 12 = 0, we’ve seen the power and universality of mathematics. Quadratic equations, like ours, tell stories of intersections and solutions, revealing the intricacies hidden within the numbers and symbols. We successfully decoded this story using the quadratic formula and uncovered our solutions. So while such equations may initially seem daunting, once the underlying principles are understood, they become a fascinating gateway into mathematics and its real-world applications.
