91Ó°ÊÓ

Roots of equations, particularly quadratic equations, are the values of the variable that satisfy the equation, meaning they make the equation true. For the quadratic function provided in the exercise, the equation is in the form \( f(x) = \frac{12-x^2}{5} \). To find the roots, we set \( f(x) = 0 \). This leads to solving the equation:

• \( \frac{12-x^2}{5} = 0 \)

Multiplying both sides by 5 simplifies it to \( 12 - x^2 = 0 \). Further simplification gives \( x^2 = 12 \). The roots are found by taking the square root of both sides, resulting in:

• \( x = \pm 2\sqrt{3} \)

Thus, the solutions \(-2\sqrt{3}\) and \(2\sqrt{3}\) are the points where the graph of the quadratic equation crosses the x-axis. These points are crucial as they provide a clear picture of the behavior of the function, and knowing how to find them is an essential skill in algebra.

The vertex of a parabola is a significant feature of a quadratic function, as it represents either the highest or lowest point on the graph, depending on the direction the parabola opens. For the equation \( f(x) = \frac{12-x^2}{5} \), we are dealing with a quadratic function that opens downward since the leading coefficient, \(a = -1\), is negative.

• The formula for finding the vertex of a quadratic function \( f(x) = ax^2 + bx + c \) is given by \((-b/2a, f(-b/2a))\).
• In our case, where \(b = 0\), the vertex simplifies to \( (0, f(0)) \).

By substituting \( x = 0 \) into the function, the vertex is calculated as follows:

• \( f(0) = \frac{12 - 0^2}{5} = \frac{12}{5} \)
• Thus, the vertex is \( (0, \frac{12}{5}) \)

Understanding the vertex is vital, as it helps in sketching the graph and analyzing the function's maximum or minimum values.

Solving equations involves finding the values of the variables that make the equation true. In the context of quadratic functions, it often means finding where the function equals zero, often leading to determining the roots. The solving process typically follows these steps:

• Set the function equal to zero: \( f(x) = 0 \).
• Restructure the equation to isolate terms, for instance, \( \frac{12-x^2}{5} = 0 \) becomes \( 12 - x^2 = 0 \) after multiplying by 5.
• Solve the resulting equation by algebraic manipulation: \( x^2 = 12 \).
• Find the values of \(x\) by taking square roots: \( x = \pm 2\sqrt{3} \).

This simple process highlights an essential algebraic skill that can widely apply to various mathematical problems, assisting students in developing a stronger grasp of mathematics. It's also a gateway to understanding more complex equations and functions.