Question

14. Arătaţi că următoarele expresii sunt pătrate ale unor numere reale, oricare ar fi x = R: b)n= x(x−5)(r – 5x+8)+16; d) n = (x² - 2x + 3)(x² - 2x-1) + 4; f) n = (x²-x)(x²-x-6) +9; h) n = (x² + x - 1)(x²+x-11) + 25. a) n = x(x − 3)(x² − 3x + 4) + 4; - c) n = (x²-x+ 2)(x²-x+8)+9; e) n = (x² + x)(x²+x+10) + 25; g) n = (x² + 3x)(x² + 3x - 8) + 16;​

Answer 1

Răspuns:

Explicație pas cu pas:

a) n = x(x-3)(x²-3x+4)+4 = (x²-3x)(x²-3x+4)+4 = a(a+4)+4 = a²+4a+4 =

= (a+2)² = (x²-3x+2)²

--------

b) n = x(x-5)(x²-5x+8)+16 = (x²-5x)(x²-5x+8)+16 =

= a(a+8)+16 = a²+8a+16 = (a+4)² = (x²-5x+4)²

--------------

c) n = (x²-x+2)(x²-x+8)+9 = (a+2)(a+8)+9 = a²+10a+25 = (a+5)²=

= (x²-x+5)²

------------

d) n = (x²-2x+3)(x²-2x-1)+4 = (a+3)(a-1)+4 = a²+2a+1 = (a+1)² =

= (x²-2x+1)² = (x-1)⁴

------------

e) n = (x²+x)(x²+x+10)+25 = a(a+10)+25 = a²+10a+25 = (a+5)²=

= (x²+x+5)²

--------------

f) n = (x²-x)(x²-x-6)+9 = a(a-6)+9 = a²-6a+9 = (a-3)²=

= (x²-x-3)²

-------------

g) n = (x²+3x)(x²+3x-8)+16 = a(a-8)+16 = a²-8a+16 = (a-4)²=

= (x²+3x-4)²

------------

h) n = (x²+x-1)(x²+x-11)+25 = (a-1)(a-11)+25 = a²-12a+36 = (a-6)²=

= (x²+x-6)²