Question

Fie expresia E(x)=(x+1)² +2(x-7)+1
a)aratati ca E(x)=(x-2)(x+6)

Answer 1

 E(x)=(x+1)² +2(x-7)+1

(x+1)² +2(x-7)+1 = 0 => x² +2x+1+2x-14+1 = 0 =>x² +4x-12 = 0

Δ = 16+48 =64 => 

$x_{1,2} = \frac{-4\pm8}{2} \Rightarrow \left \{ {{x_1=2} \atop {x_2=-6}} \right. \\ \\ \ Stim ca la o ecuatie de gradul 2: ax^2+bx+c = 0 \\ ax^2 +bx+c poate fi scrisa a(x-x_1) (x-x_2)\\ \\ la noi ecuatia este x^2+4x-12 = 0 \Rightarrow \\ \\ \Rightarrow E(x) = x^2+4x-12 = 1\cdot(x-2)(x-(-6)) \Rightarrow \\ \Rightarrow E(x) = (x-2)(x+6)$

Answer 2

E(x)=(x+1)²+2(x-7)+1
[Folosim regula de calcul prescurtat (a+b)²=a²+2ab+b²]
E(x)=(x²+2•x•1+1²)+2x-14+1
E(x)=x²+2x+1+2x-13
E(x)=x²+4x-12
[Suma si produs]
E(x)=x²-2x+6x-12
E(x)=x(x-2)+6(x-2)
E(x)=(x-2)(x+6)