Question

15. Докажите тождества:
a) (a + b)2 + (a - b)2 = 2(a² + b²);
c) a² + b² = (a + b)² - 2ab;
e) (2a + b)² + b² = 2(a² + (a + b)2);
b) (a + b)² - (a – b)² = 4ab;
d) a² + b2 = (a – b)2 + 2ab.
f) (ax - ay)² = a²(x - y)².​

Answer 1

a) (a + b)² + (a - b)² = a² + 2ab + b² + a² - 2ab + b² = 2(a² + b²)

c) (a + b)² - 2ab = a² + 2ab + b² - 2ab = a² + b²

e) (2a + b)² + b² = 4a² + 4ab + b² + b² = 2a² + 2(a + b)²

b) (a + b)² - (a - b)² = a² + 2ab + b² - (a² - 2ab + b²) = 4ab

d) (a - b)² + 2ab = a² - 2ab + b² + 2ab = a² + b²

f) (ax - ay)² = a²(x^2 - 2xy + y^2) = a²(x - y)²