Quadratic equations are fundamental in mathematics, frequently encountered in various fields such as physics, engineering, and economics. One such quadratic equation is 4x^2 - 5x - 12. In this article, we'll delve into what this equation represents, how to solve it, and its significance in real-world applications.

Breaking Down the Equation:

The quadratic equation 4x^2 - 5x - 12 is in the standard form: ax^2 + bx + c, where:

a = 4

b = -5

c = -12

This equation represents a quadratic polynomial, which is a second-degree polynomial. The term "quadratic" comes from the Latin word "quadratus," meaning "square," indicating the presence of the squared term (x^2).

Solutions of the Quadratic Equation:

To find the solutions of this equation, we can use various methods such as factoring, completing the square, or using the quadratic formula. The quadratic formula states that for any quadratic equation in the form ax^2 + bx + c = 0, the solutions (also known as roots) are given by:

�=−�±�2−4��2�x=2a−b±b2−4ac​​

Plugging in the values from our equation, we get:

�=−(−5)±(−5)2−4(4)(−12)2(4)x=2(4)−(−5)±(−5)2−4(4)(−12)​​

�=5±25+1928x=85±25+192​​

�=5±2178x=85±217​​

Thus, the solutions are:

�=5+2178x=85+217​​ and �=5−2178x=85−217​​

Graphical Representation:

When plotted on a graph with x on the horizontal axis and y on the vertical axis, the quadratic equation 4x^2 - 5x - 12 represents a parabola. The vertex of the parabola can be found using the formula:

�=−�2�x=2a−b​

For our equation:

�=−(−5)2(4)=58x=2(4)−(−5)​=85​

To find the y-coordinate of the vertex, we substitute �=58x=85​ into the original equation:

�=4(58)2−5(58)−12y=4(85​)2−5(85​)−12

�=258−258−12y=825​−825​−12

�=−12y=−12

So, the vertex is at (58,−12)(85​,−12).

Real-world Applications:

Quadratic equations are pervasive in various fields. In physics, they describe the trajectory of projectiles or the motion of objects under the influence of gravity. In engineering, they model the behavior of mechanical systems such as springs and pendulums. In economics, they are used to analyze profit maximization and cost minimization in business.

In conclusion, the quadratic equation 4x^2 - 5x - 12 represents a fundamental mathematical concept with diverse applications. Understanding its solutions and graphical representation provides insights not only into algebra but also into its real-world implications.