What are the roots of the quadratic function?
The roots (or zeros) of the quadratic function are the points where the graph intercepts the x-axis. To calculate the roots, it is necessary to write the function equal to zero, obtaining a second degree polynomial equation. There are many methods to solve second degree polynomial equations but, the most used one is the application of the Bhaskara formula, also known as the quadratic formula:
So, using the formula to solve the equation ax² + bx + c = 0, we get:
The graph of a quadratic function is a line known as a parabola and it intercepts the x-axis in, up to, two points:
- if b2-4ac > 0, the equation has two distinct real roots and the parabola intercepts the x-axis in two different points (x1 ≠ x2);
- if b2-4ac = 0, the equation has two equal real roots and the parabola is tangent to the x-axis (x1 = x2);
- if b2-4ac < 0, the equation has no real roots and the parabola does not intercept the x-axis;
Example of how to calculate the roots of a quadratic equation
Find the zeros of the function f(x) = x2 – 5x + 6.
a = 1
b = – 5
c = 6
x2 – 5x + 6 = 0
Replacing in the formula:
The value of b2-4ac > 0 , so there are two real distinct roots, which are 2 and 3, meaning that the function intercepts the x-axis at the points (2, 0) and (3, 0).