The properties of a quadratic equation refer to the characteristics or attributes that describe the equation. One of the key symmetric properties of a quadratic equation is its axis of symmetry. Here are the main symmetric properties of a quadratic equation:

1. Axis of Symmetry (Line of Symmetry): The axis of symmetry of a quadratic equation is a vertical line that divides the parabolic graph of the equation into two symmetrical halves. This line passes through the vertex of the parabola, which is the highest or lowest point on the graph, depending on whether the parabola opens upward or downward. The equation of the axis of symmetry can be represented as x = -b / (2a) in the standard form of a quadratic equation, ax^2 + bx + c = 0.
2. Vertex Symmetry: The vertex of a quadratic equation represents the point of maximum or minimum value of the quadratic function, depending on whether the parabola opens upward or downward. The axis of symmetry passes through the vertex, making the equation symmetric with respect to this point.
3. Root Symmetry: If a quadratic equation has real roots (solutions), it exhibits another form of symmetry. The two real roots are equidistant from the axis of symmetry. This means that if you measure the distance from the axis of symmetry to each of the roots, you’ll find that they are equal. This property is also known as the “zero property” because the quadratic function equals zero at the roots.
4. Parity Symmetry: Quadratic equations also exhibit a form of symmetry based on the parity (even or odd) of the coefficients. Specifically, if the coefficients of the quadratic equation are such that a = c (the coefficient of the x^2 term is equal to the constant term), then the graph of the equation is symmetric about the y-axis. This means that if you reflect one half of the parabolic graph across the y-axis, you’ll get the other half.
5. Vertex Symmetry Formula: The formula for finding the x-coordinate of the vertex of a quadratic equation is x = -b / (2a). This formula also represents the equation of the axis of symmetry. When you substitute this value back into the original quadratic equation, you can find the corresponding y-coordinate of the vertex.

In summary, the axis of symmetry is a fundamental symmetric property of a quadratic equation, and it divides the graph into two symmetrical halves. Additionally, quadratic equations may exhibit other forms of symmetry depending on the coefficients and roots, such as vertex symmetry and root symmetry. These properties provide valuable insights into the behavior of quadratic functions and their graphs.