Answer:
The x-intercepts of the function are [tex]x = 3[/tex] and [tex]x = -5[/tex].
Step-by-step explanation:
We are given the function [tex]\displaystyle f(x) = x^2 + 2x - 15[/tex].
In order to find the x-intercepts of the equation, we need to solve this quadratic. We can use factoring methods to solve it.
The parent function for a quadratic is [tex]f(x) = ax^2 + bx + c = 0[/tex].
We can use this information to determine what factoring technique we want to use.
There are several different methods, including using the quadratic formula, but because this is easily factorable, we can avoid using it.
We see that our a is equal to 1 (there is no coefficient in front of x²), so we can use the easier of the techniques:
Therefore, let's determine what our c is: 15.
Now, let's label the factors for 15.
Therefore, our two factors are 3 and 5 since they cannot be broken down any further (they are prime numbers - we just performed prime factorization).
Now, we need to pay attention to the sign for c. It's negative, so one of our factors needs to be negative. Therefore, we can test this out:
Now, we check for which one matches the sign for b. This is positive 2, so our two factors are 5 and -3.
Finally, to factor, we add or subtract the factors from x depending on the sign. If the factor is positive, we add it. If the factor is negative, we subtract it. Therefore, we can set up our equations.
[tex]x + 5 = 0\\\\x = -5[/tex]
[tex]x - 3 = 0\\\\x = 3[/tex]
Therefore, for when x = 3 or x = -5, the line crosses the x-axis. These are the x-intercepts of the function.
Our answers are x = -5 and x = 3.
