How to calculate x intercept?

Table of Contents

What is x intercept?

Before we explain what is x intercept, we should see where we find this term.  Foremost, we need to know what the quadrants are.  

Quadrants are called the four (4) parts that separate the plane from the vertical axes x and y.  We can see their name and the signs of the coordinates of their points in the figure below.

Graph #1

Εach point in the diagram above is defined by a specific value of the x-axis and a specific value of the y-axis and we call them coordinates of a point (x, y).  So, the coordinates of a point are a pair of ordered numbers that determine the position of the point in the Cartesian plane.  Specifically, x shows how much right or left we move from the point 0 (0,0) and y shows how up or down we move from 0 (0,0).

For example:  

Graph #2

At this point it is important to see what we call a function.  

Suppose we have a set of values of x which we will call M and let’s say that we also have a set of y values which we will call P.  The function is the process by which each x coordinate of the set M is assigned to a coordinate y of the set P.  Therefore, to define a function, we need the set of values of x, which is also the field of definition, the set of y values and a formula that will describe it.

A function’s graph is the set of all points x and y whose coordinates verify the type of function.

For example, we have the equation y=3x and the point A (3,6) belongs to its graph because if we substitute x and y, we get:  y =3x => 6 =3*3 => 6 = 6.

If we are given a point B (3,8) we can find that it does not belong to its graph because, 8 =3*3 => 8 6

Graph #3

Now, having seen and understood the above we can go back to the question, what is x intercept and answer it very easily.  

The x intercept is the value of the x coordinate of a point where any type of linear function intersects the x-axis.

For example, if we get a better look of graph #3, we can see that the x intercept of y = 3x equation is 0.  In fact, all the values on x-axis could be an x coordinate of a point.  Then we would write it as: (-2,0), (-1,0), (0,0), (1,0), (2,0) etc. 

X intercept definition.  

One definition that will give us the full description and explanation of the x intercept is:

A linear equation can pass over the x-axis and intersect it.  The point of the intersection or the value x of that point is called x intercept and for each value of x, the value of y equals zero.

X intercept formula. 

The general formula which is widely used to find the x intercept is a line and it is formulated in the following way: 

y = ax + b

As we have learned so far, the x and y are the two needed coordinates of a point on a line and both of them are unknown to us.  But, when a point is on the x-axis, we recognize at once that y coordinate will be equal to zero (0) and then we have only one unknown, which is the x intercept.  So, it is easy now to find the x intercept by setting y = 0 and then to solve the given equation for x.  Once we find the unknown x, we get the value of the x intercept and we have the needed information about the point which crosses the x-axis.  The last step is to write the coordinates of the point with the known form (x,y). Another information is that a is the slope of the line. 

For example:  

Assuming that we are given the equation of a line, y = 2x + 10 and we are asked in the exercise to find the x intercept.  We just follow the steps as we saw above.

Step 1:  We set y = 0.

Step 2:  We substitute with 0 so the equation will be like this:   0 = 2x + 10

Step 3:  We solve for x.  

0 = 2x + 10 => -10 = 2x => x = - \frac{10}{2} \Rightarrow x = -5

Step 4:  To write the two coordinates with the right form.  Once we set y = 0, we knew that the point was 

(x,0).  We found that x = -5, so the point will be written as (-5,0).

We may be given various forms of equations to find the x intercept but in any case, we follow the above simple steps.  The only thing we should always remember is that when we are asked to find the x intercept, automatically y equals to zero.

How to find x intercept.  

With the previous example we already saw one of the ways we can calculate the x intercept.  Each way is differentiated depending on the data of the exercise.  Let’s see what difference we might see in an exercise.


Τhe general formula we saw above is not the only one. We can use many more depending on the data given to us in each exercise.

Some examples are:

Example #1.  

y = ax

Once we see this kind of equation, we do not have to do anything in order to calculate x intercept. 


Because if we set y = 0 and substitute in the equation, we will find:  0 = ax \Rightarrow \frac{0}{a} = x \Rightarrow x = 0

Therefore, whatever a value has, we find x always equal to zero and the point will be 0 (0,0).

Example #2.

ax2 + bx + c = 0

In this equation we see that x is raised to the force 2 and we call it 2nd degree equation.  It is a kind of equation that gives us two solutions for x, so we can find two x intercepts.  Now we do not have just a line, but we have a parabola.


Let’s say that we are given the equation:  4x^2 +32x+60=0{}   and we have to find the x intercept.  

We will use the quadratic equation:  x=\frac{-b\pm \sqrt{b^2{-4ac}}}{2a}\Rightarrow x=\frac{-32b\pm \sqrt{32^{2}-4*4*60}}{2*4}\Rightarrow x=\frac{-32\pm \sqrt{1.024-960}}{8}

We have to solve two formulas because of \pm

So, x_{1} ={}\frac{-32+\sqrt{64}}{2a}\Rightarrow x=\frac{-32+8}{8} \Rightarrow x =\frac{-24}{8}\Rightarrow x=-3

and x_{2} ={}\frac{-32-\sqrt{64}}{2a}\Rightarrow x=\frac{-32-8}{8} \Rightarrow x =\frac{-40}{8}\Rightarrow x=-5

We just found the two values of x intercept so the two points which are crossing the x-axis are:

(-3,0) and (-5,0)


When we are given a graph, we can see very easily where the line intersects the x-axis.  Usually in these exercises we are asked to find the equation.  

Example #1.

We are given the following graph and we have to find the form of equation.  

We first have to find the point where the line is on the x-axis.  If we take a good look, we will see that the line crosses the point 0 (0,0).  What does that tell us?  We saw in another case that when we have a point 0(0.0), we have the equation of the form y = ax.

Example #2.

We are given the following graph and we have to find the x intercept.  

We can easily see that the point of the line which crosses the x-axis is (2,0), so the x intercept is 2.

Summing up, it is easy to find x intercept, as long as we remember that in this case that x intercept is always found on the x-axis and that the coordinate y is always equal to zero.  So, we are left with only one unknown and we can solve the equation in relation to x.  We do the same when we are asked to find the y intercept.

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