If you are looking at a graph of a quadratic equation, how do you determine where the solutions are?
If you are looking at the graph of a
Y- parabola, the real solutions are
the point(s) where the parabola touches
or crosses the X-axis. If there is only
one real solution, the vertex will be
the only point to touch the X-axis.
If no point on the parabola touches or
crosses the X-axis, there are 2 imaginary solutions and no real
solutions.
To determine where the solutions of a quadratic equation lie, you can examine the graph of the equation. There are two main methods to do this:
1. Graphical method: Plot the quadratic equation on a coordinate plane to create its graph. The solutions, or roots, of the equation correspond to the x-intercepts of the graph. Locate the points where the graph intersects or touches the x-axis. These points indicate where the solutions are located.
2. Algebraic method: The solutions of a quadratic equation can also be determined algebraically using the quadratic formula or factoring. The quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), helps find the x-values corresponding to the solutions. If the discriminant (the value inside the square root, b^2 - 4ac) is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
By graphing the quadratic equation or solving it algebraically, you can determine where the solutions lie on the graph.