Quadratic equations are an important topic in algebra and mathematics. They are used to model a variety of real-world problems, from projectile motion to financial analysis. One common question that arises when studying quadratic equations is whether a given sequence is a quadratic equation. In this article, we will explore the sequence 4x^2 - 5x - 12 = 0 and determine whether it is a quadratic equation. We will discuss the characteristics of quadratic equations and provide a step-by-step analysis of the given sequence to determine its classification. 

Characteristics of quadratic equations 

Quadratic equations are polynomial equations of degree two. They have the general form of ax^2 + bx + c = 0, where a, b, and c are coefficients. The coefficient of x^2 (a) must be non-zero for an equation to be quadratic.

The discriminant (b^2 - 4ac) is a key characteristic of quadratic equations. It determines the nature of the roots of the equation. If the discriminant is positive, the roots are real and distinct. If it is zero, the roots are real and equal. If it is negative, the roots are complex conjugates.

Is the sequence 4x ^ 2 - 5x - 12 = 0?

Now let's analyze the given sequence, 4x^2 - 5x - 12 = 0. We can see that it is indeed a second-degree polynomial equation with the highest power of x being 2. The coefficients of x^2, x, and the constant term are 4, -5, and -12, respectively. Therefore, we can conclude that the given sequence is indeed a quadratic equation.

To determine the nature of the roots of this equation, we need to calculate the discriminant. The discriminant is b^2 - 4ac = (-5)^2 - 4(4)(-12) = 25 + 192 = 217. Since the discriminant is positive, the roots are real and distinct.

To solve for the roots, we can use the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a. Plugging in the values of a, b, and c from our given sequence, we get:

x = (5 ± sqrt(217)) / 8

Therefore, the solutions to the quadratic equation 4x^2 - 5x - 12 = 0 are x = -3/4 and x = 4/3.

In conclusion, we have determined that the given sequence 4x^2 - 5x - 12 = 0 is a quadratic equation with real and distinct roots. By calculating the discriminant, we were able to determine the nature of the roots, and by using the quadratic formula, we were able to solve for the values of x.