Question

33. Se considera expresia E(x) =(2x+3)^2-(x-3)(2+x)-(x+2)^2-x(x+7)-10,unde x este un numar real
a)Aratati ca E(x) = (x+1)^2 ,pentru orice numar real x
b)Determinati numarul intreg a , pentru care E (a^2) - E(-a^2)=16

Answer 1

Explicație pas cu pas:

a)

$E(x) = (2x+3)^{2} - (x-3)(2+x) - (x+2)^{2} - x(x+7) - 10 =$

$= 4 {x}^{2} + 12x + 9 - (2x + {x}^{2} - 6 - 3x) - ( {x}^{2} + 4x + 4) - ( {x}^{2} + 7x) - 10$

$= 4 {x}^{2} + 12x + 9 - {x}^{2} + 6 + x - {x}^{2} - 4x - 4 - {x}^{2} - 7x - 10$

$= (4 {x}^{2} - 3 {x}^{2}) + (13x - 11x) + (15 - 14)$

$= {x}^{2} + 2x + 1 = \bf {(x + 1)}^{2}$

b)

$E (a^{2}) - E(-a^{2}) = 16$

${( {a}^{2} + 1)}^{2} - {( - {a}^{2} + 1)}^{2} = 16$

$[ ({a}^{2} + 1) + (-{a}^{2} + 1)][ ({a}^{2} + 1) - (-{a}^{2} + 1)] = 16$

$({a}^{2} + 1 - {a}^{2} + 1)({a}^{2} + 1 + {a}^{2} - 1) = 16$

$2 \cdot 2{a}^{2} = 16 \iff 4{a}^{2} = 16$

${a}^{2} = 4 \iff {a}^{2} = {2}^{2}$

$a \in \{-2; 2\}$