Solution of quadratic equation using graph

Based on the sources and our conversation history, we can explain how to solve a quadratic equation using the graphing method.

First, let’s understand the connection between quadratic equations and quadratic functions. A quadratic equation is formed when a quadratic function, defined as f(x) = ax² + bx + c where a ≠ 0, is set equal to a specific value. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are integers and a is not equal to 0. The solutions to this equation are called the roots of the equation.

The core idea behind solving a quadratic equation using its graph lies in the relationship between the roots of the equation ax² + bx + c = 0 and the zeros of the associated quadratic function f(x) = ax² + bx + c. The zeros of the function are the x-values for which f(x) = 0. Graphically, these zeros correspond precisely to the x-intercepts of the parabola that represents the quadratic function.

Therefore, to solve a quadratic equation ax² + bx + c = 0 using the graphing method, you can follow these steps:

1. Identify the associated quadratic function: This is f(x) = ax² + bx + c.
2. Plot the graph of the quadratic function: This involves drawing the parabola. To do this effectively, the sources suggest a “cookbook recipe”:
   • Find the axis of symmetry, which is given by the equation x = -b/(2a). This value depends only on the coefficients ‘a’ and ‘b’.
   • Find the y-intercept by setting x = 0 in the function, which gives f(0) = c.
   • Find the vertex of the parabola. The x-coordinate of the vertex is the same as the axis of symmetry, x = -b/(2a). The y-coordinate of the vertex is found by substituting this x-value back into the function, f(-b/(2a)). This y-coordinate represents the minimum or maximum value of the function.
   • Plot these key points and potentially others around the axis of symmetry, using the symmetry property, to sketch the smooth curve of the parabola.
3. Identify the x-intercepts: Once the parabola is plotted, look for the points where it crosses or touches the x-axis.
4. Determine the roots: The x-coordinates of these x-intercepts are the roots of the quadratic equation ax² + bx + c = 0.

The number of real roots a quadratic equation has can be determined by how many times the graph of the associated quadratic function intersects the x-axis:

• Two real roots: The parabola crosses the x-axis at two distinct points. This occurs when ‘a’ > 0 and the minimum value (y-coordinate of vertex) is negative, or when ‘a’ < 0 and the maximum value (y-coordinate of vertex) is positive.
• One real root (repeated): The parabola touches the x-axis at exactly one point, which is the vertex. This happens when the minimum or maximum value (y-coordinate of vertex) is exactly zero.
• No real roots: The parabola does not intersect the x-axis at all. This happens when ‘a’ > 0 and the minimum value (y-coordinate of vertex) is positive, or when ‘a’ < 0 and the maximum value (y-coordinate of vertex) is negative.

While the graphing method is a good way to verify results obtained algebraically and helps visualise the solutions, it may not always give precise results, particularly if the roots are not integers.