Question

Arăți că urmatoarele expresii sunt pozitive oricare ar fi numarul real x
x^2+2x+3
x^2+6x+11
x^2+12x+39
4x^2+12x+13
9x^2+24x+19
25x^2+30x+17

Answer 1

x^2+2x+3 = x^2 + 2x + 1 + 2 = (x + 1) ^2 + 2
x^2+6x+11 = x^2 + 6x + 9 + 2 = (x + 3)^2 + 2 
x^2+12x+39 = x^2 + 12x + 36 + 3 = (x + 6)^2 + 3
4x^2+12x+13 = 4x^2 + 12x + 9 + 4 = (2x + 3)^2 + 4  
9x^2+24x+19 = 9x^2 + 24x + 16 + 3 = ( 3x + 4)^2 + 3 
25x^2+30x+17 = 25x^2 + 30x + 9 + 8 = ( 5x + 3)^2 + 8 

Answer 2

x²+2x+3=x²+2x+1+2=(x+1)²+2
Dar (x+1)²≥0 ⇒(x+1)²+2≥2 ⇔expresie pozitiva unde x∈R ;
x²+6x+11=x²+6x+9+2=(x+3)²+2
Dar (x+3)²≥0 ⇒(x+3)²+2≥2 ⇔expresie pozitiva unde x∈R;
x²+12x+39=x²+12x+36+3=(x+6)²+3
Dar (x+6)²≥0 ⇒(x+6)²+3≥3 ⇔expresie pozitiva unde x∈R;
4x²+12x+13=4x²+12x+9+4=(2x+3)²+4
Dar (2x+3)²≥0 ⇒(2x+3)²+4≥4 ⇔expresie pozitiva unde x∈R;
9x²+24x+19=9x²+24x+16+3=(3x+4)²+3
Dar (3x+4)²≥0 ⇒(3x+4)²+3≥3 ⇔expresie pozitiva unde x∈R;
25x²+30x+17=25x²+30x+9+8=(5x+3)²+8
Dar (5x+3)²≥0 ⇒(5x+3)²+8≥8 ⇔expresie pozitiva unde x∈R;