This cheat sheet provides the most frequently used formulas, with brief descriptions of what the letters and symbols represent. Counting techniques are also here, letting you count numbers of events without actually having to list all the ways to do them. Also, you find a stepbystep description of how to complete the square — most useful when you’re working with conic sections and other equations with specific formats.
Frequently used precalculus formulas
Mathematical formulas are equations that are always true. You can use them in algebra, geometry, trigonometry, and many other mathematical applications, including precalculus. Refer to these formulas when you need a quick reminder of exactly what those equations are and which measures or inputs are needed.
Counting techniques for precalculus
Counting the number of ways to perform a task is fairly simple — until the number of choices gets too large. Here are three counting techniques used in precalculus:

Permutations: How many ways you can choose r things from a total of n things when the order of your choices matters

Combinations: How many ways you can choose r things from a total of n things when the order of your choices doesn’t matter

Multiplication principle: How many different possibilities exist if you choose 1 from a things, 1 from b things, 1 from c things, 1 from d things, and so on.
Use the following formulas for these counting techniques:
Completing the square for conic sections
When the equation of a conic section isn’t written in its standard form, completing the square is the only way to convert the equation to its standard form. The steps of the process are as follows:

Add/subtract any constant to the opposite side of the given equation, away from all the variables.

Factor the leading coefficient out of all terms in front of the set of parentheses.

Divide the remaining linear coefficient by two, but only in your head.

Square the answer from Step 3 and add that inside the parentheses.
Don’t forget that if you have a coefficient from Step 2, you must multiply the coefficient by the number you get in this step and add that to both sides.

Factor the quadratic polynomial as a perfect square trinomial.