**The Equation – 4x ^ 2 – 5x – 12 = 0 :- **Quadratic equations are fundamental in algebra and play a crucial role in different areas of mathematics and science. Take the equation (**– 4x ^ 2 – 5x – 12 = 0)** as an example; it follows the standard form of a quadratic equation, (ax^2 + bx + c = 0), where (a), (b), and (c) are constants, and (x) is the variable. The uniqueness of this equation lies in its coefficients and the solutions it produces, making it an interesting case to study.

**Method of Solution 4x ^ 2 – 5x – 12 = 0**

Solving quadratic equations offers various methods, each bringing its own perspective and usefulness. Among the commonly used techniques are factorization, completing the square, and the quadratic formula.

Factorization works by expressing the quadratic as the product of two binomials, while completing the square involves transforming the equation into a perfect square form. However, the quadratic formula stands out as the most widely applicable method, especially for equations that prove challenging to factorize.

Certainly! Let’s solve the quadratic equation (** -4x^2 – 5x – 12 = 0** ) step by step using the quadratic formula:

[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} ]

where ( a ), ( b ), and ( c ) are the coefficients of the quadratic equation ( ax^2 + bx + c = 0 ).

In your equation, ( a = -4 ), ( b = -5 ), and ( c = -12 ). Plug these values into the quadratic formula:

[ x = \frac{-(-5) \pm \sqrt{(-5)^2 – 4(-4)(-12)}}{2(-4)} ]

**Simplify the expression:**

[ x = \frac{5 \pm \sqrt{25 – 192}}{-8} ]

[ x = \frac{5 \pm \sqrt{-167}}{-8} ]

Now, the expression inside the square root (( -167 )) is negative, which means that the solutions will involve imaginary numbers. To simplify the expression, we can factor out ( i ), the imaginary unit (( \sqrt{-1} )):

[ x = \frac{5 \pm i\sqrt{167}}{-8} ]

So, the solutions to the quadratic equation ( -4x^2 – 5x – 12 = 0 ) are:

[ x = \frac{5 + i\sqrt{167}}{-8} ]

and

[ x = \frac{5 – i\sqrt{167}}{-8} ]

These are the complex roots of the quadratic equation. If you have any further questions or need clarification on any step, feel free to ask!

**Significance and Applications** – (Explanation)

Quadratic equations play a crucial role in many areas of science and engineering. Think of them like powerful tools that help us understand and solve real-world problems. For example, the roots of these equations can stand for things we care about, like how long something takes to fly or the highest point it reaches in projectile motion.

They’re not just for physics; they also come in handy in economics and chemistry. In economics, quadratic equations help find the best solutions, like the ideal price for maximum profit. In chemistry, they point to where reactions reach a stable state.

When we dig into these equations and figure out if the solutions are real, complex, or repeated, it gives us a peek into how these systems behave. It’s like unraveling the secrets of the world around us using the language of math.

**Real-World Example**

In physics, the trajectory of a projectile is often modeled by a quadratic equation, where the solutions can indicate the time at which the projectile reaches a certain height.

Similarly, in economics, the profit maximization problem can be modeled as a quadratic equation, where the roots indicate maximum or minimum profit points.

**Conclusion** of (– 4x ^ 2 – 5x – 12 = 0)

The equation** (– 4x ^ 2 – 5x – 12 = 0) **isn’t just a bunch of math symbols; it’s like a key that unlocks insights into the natural and social sciences. When we explore the solutions to this equation, we discover important patterns and principles that are essential in different areas of study. It shows us how closely connected math is to the world we live in, revealing a deeper understanding of complex phenomena. In simpler terms, this equation helps us see the big picture and how math plays a crucial role in explaining things around us.

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