how to find x intercept of quadratic function

Given the graph of any role, an x-intercept is simply the signal, or points where the graph crosses the x-centrality. In that location might exist just ane such indicate, no such indicate, or many, significant a part can have several ten-intercepts. As yous will encounter beneath, we tin utilize a graph or a simple algebra dominion to find the x-intercept or 10-intercepts of whatsoever role. You can as well ringlet downwards to a video example below.

Table of Contents

Using a graph to notice x-intercepts
Using algebra to notice x-intercepts
Video instance (including when there are no 10-intercepts)
Further reading

Finding the x-intercept or x-intercepts using a graph

Equally mentioned above, functions may have one, zilch, or even many x-intercepts. These can be institute by looking at where the graph of a function crosses the x-axis, which is the horizontal axis in the xy-coordinate plane. You can see this on the graph below. This function has a unmarried x-intercept.

a graph with an x-intercept of 2

In the graph beneath, the office has two ten-intercepts. Discover that the course of the point is always \((c, 0)\) for some number \(c\).

Graph showing a function with two x-intercepts. The graph crosses the x-axis two times.

Finally, the following graph shows a function with no x-intercepts. You tin can see this because information technology does not cross the x-axis at any point.

Graph showing a function with no x-intercepts. The function's graph does not cross the x-axis.

Y'all can meet a more advanced discussion of these ideas here: The zeros of a polynomial.

Finding the x-intercept or intercepts using algebra

The general rule for finding the x-intercept or intercepts of any role is to let \(y = 0\) and solve for \(10\). This may be somewhat easy or actually difficult, depending on the function. Let's look at some examples to see why this may be the case.

Example

Find the ten-intercept of the office: \(y = 3x – nine\)

Solution

Allow \(y = 0\) and solve for \(x\).

\(\begin{align}0 &= 3x – 9\\ -3x &= -9\\x &= three\cease{align}\)

Answer: Therefore the x-intercept is three. You lot could also write it as a point: \((three,0)\)

A more than complicated example would be 1 where the equation representing the function itself is more complex. For these situations, you need to know a picayune more algebra in order to discover any intercepts.

Example

Find the x-intercepts for the function: \(y = x^2 + 2x – 8\)

Solution

As before, let \(y = 0\) and solve for \(10\). This time, you take a quadratic equation to solve.

\(\begin{align} 0 &= x^2 + 2x – viii\\ 0 &= (x + 4)(ten – two)\\ x &= -4, 2\end{align}\)

Answer: This function has two x-intercepts: –4 and 2. These are located at \((–4, 0)\) and \((2, 0)\).

For equations more than complex than this, a graphing calculator is often useful for at least estimating the location of any intercepts.

Video examples

In the following video, y'all tin can run into how to find the x-intercepts of 3 different functions. This likewise includes an instance where there are no x-intercepts.