91Ó°ÊÓ

Quadratic equations are a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations are called quadratic because the highest power of the variable \(x\) is squared (\(x^2\)). Quadratic equations are essential because they describe parabolic relationships which appear in various contexts such as physics, biology, and economics. Solving a quadratic equation means finding the values of \(x\) that satisfy the equation. There are several methods to solve quadratics, including:

• Factoring: Expressing the quadratic in a product of two binomials.
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
• Completing the square: Rewriting the equation in the form \((x-h)^2=k\).
• Graphing: Finding where the parabola intersects the x-axis.

In business applications, quadratic equations help in modeling profit, cost, and revenue functions, making them vital for maximizing profit.

The vertex formula is crucial for finding the highest or lowest point on the graph of a quadratic equation (the parabola). The vertex represents a maximum or minimum point depending on the orientation of the parabola. For a quadratic equation in the form \(ax^2 + bx + c\), the vertex point \((h, k)\) occurs at:

• \(h = -\frac{b}{2a}\)
• Substituting \(x = h\) into the equation gives \(k\).

The vertex formula simplifies the process of identifying profit maximization or minimization points in business applications. For example, in our café problem, the quadratic function \(P = -0.5x^2 + 4x + 120\) has its vertex at \(x = -\frac{4}{2(-0.5)} = 4\). Thus, adding 4 additional tables to the initial 12 tables (total 16 tables) will maximize the profit.
Using the vertex formula thus provides a quick and reliable method to find the optimal conditions for various scenarios modeled by quadratic equations.

Business mathematics involves the application of mathematical methods and techniques in business and commerce to optimize creative strategies, operations, and financial outcomes. One key area is modeling profit functions and understanding how changes in variables affect profit. For example, in our café problem, we:

• Defined variables: setting \(x\) as additional tables beyond the initial 12.
• Formulated a profit equation \(P = (12 + x)(10 - 0.5x)\) based on given conditions.
• Simplified the profit equation into a quadratic function \(P = -0.5x^2 + 4x + 120\).

Using business mathematics, we transformed real-world constraints into a solvable problem, where the optimal number of additional tables that maximize profit was found using quadratic equation approaches. Through these mathematical tools, businesses can make data-driven decisions ensuring efficiency, growth, and profitability. Mastering these fundamental concepts empowers individuals to tackle various business challenges systematically and effectively.