If you have taken algebra 2, you know you can write quadratic functions in three forms:
- Standard form:
- Vertex form:
- Factored form:
No matter which of the forms you have, you are often asked to find the roots (x-intercepts). The factored form is already done for you: the roots are the values of and
. If your equation is in standard form, you use the quadratic formula. But what do you do if you have a quadratic in vertex form? Most students are taught to expand the equation into standard form (and then use the quadratic formula). But there’s a quick shortcut that is pretty easy to use. In fact, you can write the roots by inspection. If
, then the roots are
There are three cases:
: There is one (double) root;
have different signs: the two roots are real;
have the same sign: the two roots are complex.
In the complex case, you can express the roots as
Here are two quick examples:
Example 1:
Using the formula above, the roots can be written immediately as
Example 2:
The roots are: