Short Multiplication Formulas

We we have a sum(difference) of two numbers and we need to remove the brakets we use the short multipication formulas:

(x + y)² = x² + 2xy + y²
(x - y)² = x² - 2xy + y²

Examples: if x = 10, y = 5a
(10 + 5a)² = 10² + 2.10.5a + (5a)² = 100 + 100a + 25a²
(10 - 4)² = 10² - 2.10.4 + 4² = 100 - 80 + 16 = 36
Of course if we have the following situation:
25 + 20a + 4a² = 5² + 2.2.5 + (2a)² = (5 + 2a)²

(x + y)³ = x³ + 3x²y + 3xy² + y³
(x - y)³ = x³ - 3x²y + 3xy² - y³

Example: (1 + a²)³ = 1³ + 3.1².a² + 3.1.(a²)² + (a²)³ = 1 + 3a² + 3a⁴ + a⁶

(x + y + z)² = x² + y² + z² + 2xy + 2xz + 2yz
(x - y - z)² = x² + y² + z² - 2xy - 2xz + 2yz

x² - y² = (x - y)(x + y)
x² + y² = there is no such a formula

Example: 9a² - 25b² = (3a)² - (5b)² = (3a - 5b)(3a + 5b)

x³ - y³ = (x - y)(x² + xy + y²)
x³ + y³ = (x + y)(x² - xy + y²)

If n is natural number

xⁿ - yⁿ = (x - y)(x^n-1 + x^n-2y +...+ y^n-2x + y^n-1)

Short Multiplication Formulas Problem

1) Solve the equation: x² - 25 = 0
Solution: x² - 25 = (x - 5)(x + 5)
=> the to solve the equation we have to solve the following 2 equations:
x - 5 = 0 or x + 5 = 0 so the equation have two decisions: x = 5 and x = -5

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