Question

1.Aratati ca E(x)= x² + x - 6 supra x² - 4 = x+3 supa x+2

2. Aratati ca 2x +6 supra x² + 4x+3 = 2 supra x+1, x∈R / ∫-3, -1,1 ∫

Answer 1

Primul exercitiu

$\frac{x^2+x-6}{x^2-4}$

[tex]x^2+x-6=0 a=1 b=1 c=-6 delta=b^2-4ac = 1^2-4*1*6=1+24=25 [/tex]

[tex]x1= \frac{-1+5}{2}= \frac{4}{2}=2 [/tex]

$x2= \frac{-1-5}{2}= \frac{-6}{2}=-3$

Conform formulei a(x-x1)(x-x2) rezulta

$x^2+x-6= (x-2)(x+3)$

$x^2-4= (x-2)(x+2)$

$\frac{(x-2)(x+3)}{(x-2)(x+2)}= \frac{x+3}{x+2}$

Exericitul 2

$\frac{2x+6}{x^2+4x+3}$

$2x+6=2(x+3)$

[tex]x^2+4x+3=0 a=1 b=4 c=3 delta=b^2-4ac = 4^2-4*1*3=16-12=4[/tex]

[tex]x1= \frac{-4+2}{2}= \frac{-2}{2}=-1 [/tex]

$x2= \frac{-4-2}{2}= \frac{-6}{2}=-3$

$x^2+4x+3= (x+1)(x+3)$


$\frac{2(x+3)}{(x+1)(x+3)}= \frac{2}{x+1}$