What are abbreviated multiplication formulas (FMS)?
Imagine you need to multiply two long parentheses, like (x + 5)(x + 5). You can do it the old fashioned way: multiply each number by each, add the results, and add similar terms. It's long and it's easy to get confused in the process.
Mathematicians have noticed that some expressions occur constantly and are always calculated according to the same scheme. In order not to waste time, these schemes were learned and called abbreviated multiplication formulas (abbreviated as FSU). These are the "cheat codes" in the algebra world that let you turn a long multiplication into a short answer in one second.
Top 3 most important formulas (Cheat Sheet)
These three formulas are the basics that everyone should know. They occur in 90% of all school problems.
- 1. Difference of squares:
a2 − b2 = (a − b)(a + b)
In simple words: The difference between the squares of two numbers is equal to the product of their difference and their sum. - 2. Squared sum:
(a + b)2 = a2 + 2ab + b2
Important: Don't forget about the “double product” (2ab) in the middle! - 3. Squared difference:
(a − b)2 = a2 − 2ab + b2
Differs from the previous one only by a minus before the two.
Summary table of all formulas
If you need more than basic squares, here's the full table, including cubes (for advanced tasks):
| Formula name | What does she look like? |
|---|---|
| Difference of squares | a2 − b2 = (a − b)(a + b) |
| Sum Square | (a + b)2 = a2 + 2ab + b2 |
| Difference square | (a − b)2 = a2 − 2ab + b2 |
| Sum of cubes | a3 + b3 = (a + b)(a2 − ab + b2) |
| Cube difference | a3 − b3 = (a − b)(a2 + ab + b2) |
⚠️ The main mistake of beginners (Caution!)
The most common mistake in tests is to write like this: (a + b)2 = a2 + b2.
This isWRONG! You have lost the middle of the expression - the double product2ab. Think of it like a recipe: if you forget one ingredient, the dish (or example) won't work. Remember: the square of the sum always gives THREE terms in the output.
Example analysis: how to use the formula?
Let's solve a real problem: simplify the expression (x + 3)2.
- We see a bracket in a square with a plus inside. So, we use the formula Square of the sum.
- In our formula, a is x, and b is 3.
- Substitute:
- First the square of the first: x2
- Then double the product: 2 × x × 3 = 6x
- At the end the square of the second: 32 = 9
Total: (x + 3)2 = x2 + 6x + 9. Quick and easy!
Where is this applied in life?
Although you are unlikely to square the sum in the store, FSU is a powerful tool in “serious” professions:
- 🏗️ Engineering and construction: Calculation of the strength of structures and bending of beams is based on quadratic functions.
- 💻 Programming (Graphics): Rendering shadows, lights and complex 3D objects in games requires constant simplification of mathematical expressions for fast operation of the video card.
- 📊 Economics: Formulas are used to calculate complex percentages and analyze risks in the markets.