# How many intercepts does #y = x^2 + x − 6 # have?

##### 1 Answer

This function has a total of three intercepts.

#### Explanation:

There are two types of intercepts a function can have,

So, starting with the

#y = (0)^2 + 0 - 6#

#y = -6#

This means that the

Now for the

#y = x^2 + x - 6 = 0#

The **discriminant** of this quadratic equation will actually tell you how many

For ageneral form quadratic equation

#color(blue)(ax^2 + bx + c = 0)#

the discriminant is defined as

#color(blue)(Delta = b^2 - 4ac)#

For your quadratic,

#Delta = 1^2 - 4 * 1 * (-6)#

#Delta = 1 + 24 = 25#

When *two disctinct real roots*, which is another way of saying that the graph of the function will intercept the **two points**.

These roots will be equal to

#color(blue)(x_(1,2) = (-b +- sqrt(Delta))/(2a)#

In your case, you have

#x_(1,2) = (-1 +- sqrt(25))/(2 * 1)#

#x_(1,2) = (-1 +- 5)/2 = {(x_1 = (-1 - 5)/2 = -3), (x_2 = (-1 + 5)/2 = 2):}#

The two

Therefore, the function will have a total of three intercepts, one

graph{x^2 + x - 6 [-20.27, 20.28, -10.14, 10.13]}