What is the connection between the solutions of the equation ax^2 + bx + c = 0 and the x intercepts of the graph of f(x) = ax^2 + bx + c?
The solution says: The solutions of the equation are the first coordinates of the x intercepts.
I did the solutions to the formula but do not understand how these are the first coordinates of the x intercepts.
Thank you.
the intercepts are points with x and y coordinates
for x-intercepts, the y coordinates are zero...but they still exist
Yes, I understand what intercepts are, but I don't understand HOW the solutions to my equation are the first coordinates of my x intercepts.
Am I thinking too much about this?
To understand the connection between the solutions of the equation and the x-intercepts of the graph, let's break it down step by step:
1. Start with the equation ax^2 + bx + c = 0. This quadratic equation represents a parabola that opens upward or downward, depending on the value of 'a'.
2. The solutions to this quadratic equation are the values of 'x' that make the entire equation equal to zero. Mathematically, we find the solutions by applying the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
3. These two solutions represent the x-coordinates of the points where the parabola intersects the x-axis, also known as the x-intercepts.
4. Graphically, the x-intercepts are the points where the parabola intersects the x-axis. The x-intercepts are represented as (x, 0), where 'x' is the x-coordinate and '0' is the y-coordinate (since the points lie on the x-axis, their y-coordinate is zero).
5. Therefore, when we find the solutions to the quadratic equation, we can treat them as the x-coordinates of the x-intercepts of the graph. The y-coordinate of each x-intercept will be zero.
In summary, the solutions of the equation ax^2 + bx + c = 0 give us the x-coordinates of the x-intercepts of the graph of f(x) = ax^2 + bx + c. The y-coordinate of each x-intercept is zero since they lie on the x-axis.