If you are looking at a graph of a quadratic equation, how do you determine where the solution are.
The solutions are where the graph intersects the x-axis, or the "zeroes".
The x-coordinates of where the graph cuts the axis form the solution set.
If the graph is tangential to the x-axis, then the roots are coincident (double roots).
If the graph does not cross the x-axis, the roots are complex.
To determine the solutions of a quadratic equation by looking at its graph, follow these steps:
1. Identify the equation: Write down the equation in the standard form, which is ax² + bx + c = 0. Here, 'a', 'b', and 'c' are constants (with 'a' not being zero).
2. Analyze the graph: Plot the graph of the quadratic equation on a coordinate plane or observe the given graph. The graph will appear as a parabola.
3. Locate the vertex: The vertex of the parabola represents the point where the graph reaches its highest or lowest point. The x-coordinate of the vertex is given by -b/2a.
4. Determine the line of symmetry: The line passing through the vertex that divides the parabola into two symmetrical halves is called the line of symmetry. It is represented by the equation x = -b/2a.
5. Find the x-intercepts: The x-intercepts, also known as roots or solutions, are the points where the parabola intersects the x-axis. To find them, set y (or f(x)) equal to zero and solve the resulting quadratic equation using either factoring, completing the square, or applying the quadratic formula.
a. Factoring: If the equation can be factored, set it equal to zero and factor it into two binomial expressions. Then, set each binomial equal to zero and solve for x.
b. Completing the square: If the equation is not easily factorable, you can convert it to vertex form by completing the square. Rewrite the equation as a perfect square trinomial, and then solve for x.
c. Quadratic formula: If factoring and completing the square are not viable options, use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Plug in the values of a, b, and c from the original quadratic equation to find the solutions.
6. Interpret the solutions: Depending on the graph, the solutions can be real, imaginary, or repeated. Real solutions are represented by x-intercepts on the graph, imaginary solutions do not intersect the x-axis, and repeated solutions occur when the parabola just touches the x-axis at a single point.
By following these steps and analyzing the graph, you can determine where the solutions of the quadratic equation lie.