What is the process to find the x-intercepts of a quadratic function?

To find the x-intercepts of a quadratic function, you first need to understand that the x-intercepts are the points where the graph of the function crosses the x-axis. This occurs when the value of the function equals zero. For a quadratic function, which typically has the form:

f(x) = ax² + bx + c

you will set the function equal to zero:

ax² + bx + c = 0

There are a few methods to solve this equation:

1. Factoring

If the quadratic can be factored, you can write it as:

(mx + n)(px + q) = 0

From here, set each factor equal to zero:

mx + n = 0 and px + q = 0

Solving each equation for x will give you the x-intercepts.

2. Using the Quadratic Formula

If factoring is difficult or impossible, you can use the quadratic formula:

x = (-b ± √(b² – 4ac)) / (2a)

This formula will yield two possible values for x, which represent the x-intercepts. Remember that the term under the square root (the discriminant b² – 4ac) determines the nature of the roots:

• If it is positive, you’ll have two distinct x-intercepts.
• If it equals zero, there is one x-intercept (the vertex of the parabola).
• If it is negative, there are no real x-intercepts (the parabola does not cross the x-axis).

3. Graphing