In the realm of mathematics, quadratic equations are a fundamental topic, and they often appear in various scientific and real-world applications. One such quadratic equation is 4x ^ 2 – 5x – 12 = 0. In this article, we’ll embark on a journey to understand and solve this particular quadratic equation, exploring the methods and principles involved.

## Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically expressed in the form ax² + bx + c = 0, where a, b, and c are coefficients. In our equation, 4x ^ 2 – 5x – 12 = 0, we have:

- a = 4
- b = -5
- c = -12

The main goal in solving a quadratic equation is to find the values of x that make the equation true.

## Using the Quadratic Formula

To solve the quadratic equation 4x ^ 2 – 5x – 12 = 0, we can use the quadratic formula, which is a powerful tool in mathematics:

x = (-b ± √(b² – 4ac)) / (2a).

Let’s apply this formula to our equation:

For our equation, a = 4, b = -5, and c = -12. Plugging these values into the formula:

x = (-(-5) ± √((-5)² – 4(4)(-12))) / (2(4)).

Now, let’s simplify this further:

x = (5 ± √(25 + 192)) / 8.

x = (5 ± √217) / 8.

We now have two possible solutions:

- x₁ = (5 + √217) / 8
- x₂ = (5 – √217) / 8.

## Interpreting the Solutions

The solutions we’ve found, x₁ and x₂, represent the x-coordinates where the graph of the quadratic equation 4x ^ 2 – 5x – 12 intersects the x-axis. In other words, these are the values of x for which the equation is true. Here’s what each solution means:

- x₁ = (5 + √217) / 8: This is one of the solutions to the equation. It represents a specific x-value where the equation 4x² – 5x – 12 = 0 is satisfied. It’s a real number, and it’s part of the solution set.
- x₂ = (5 – √217) / 8: This is the other solution to the equation. Like x₁, it represents an x-value where the equation is true. It’s also a real number and belongs to the solution set.

### Conclusion

In conclusion, the quadratic equation 4x ^ 2 – 5x – 12 = 0 has two real solutions:

- x₁ = (5 + √217) / 8
- x₂ = (5 – √217) / 8.

These solutions allow us to find the x-values where the equation is satisfied. Solving quadratic equations is a fundamental skill in algebra and mathematics, and it has numerous applications in various fields, from physics to engineering and beyond. Understanding how to use the quadratic formula is a valuable tool for problem-solving in these disciplines.