Question

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)Determina valorile numarului real a, pentru care E(a^2)-E(-a^2)=16

Answer 1

Explicație pas cu pas:

a)

$E(x) = (2x + 3)^2 - (x - 3)(2 + x) - (x + 2)^2 - x(x + 7) - 10 = (4 {x}^{2} + 12x + 9) - (2x + {x}^{2} - 6 - 3x) - ( {x}^{2} + 4x + 4) - {x}^{2} - 7x - 10 = 4 {x}^{2} + 12x + 9 - {x}^{2} + x + 6 - {x}^{2} - 4x - 4 - {x}^{2} - 7x - 10 = (4 {x}^{2} - 3{x}^{2}) + (13x - 11x) + (15 - 14) = {x}^{2} + 2x + 1 = \bf {(x + 1)}^{2}$

b)

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

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

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

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

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

${a}^{2} = 4 \iff |a| = 2 \implies \bf a = \pm 2$

$S = \Big\{-2 ; 2\Big\}$

Answer 2

$\displaystyle a)~E(x)=(2x+3)^2-(x-3)(2+x)-(x+2)^2-x(x+7)-10\\\\ E(x)=(2x)^2+2 \cdot 2x \cdot 3+3^2-\Big(2x+x^2-6-3x\Big)-\Big(x^2+2\cdot x \cdot 2+2^2\Big)-\\\\-x^2-7x-10\\\\ E(x)=4x^2+12x+9-2x-x^2+6+3x-x^2-4x-4-x^2-7x-10\\\\ E(x)=x^2+2x+1\\\\ E(x)=x^2+x+x+1\\\\ E(x)=x(x+1)+1(x+1)\\\\ E(x)=(x+1)(x+1)\\\\ E(x)=(x+1)^2$

$\displaystyle b)~E\Big(a^2\Big)-E\Big(-a^2\Big)=16\\\\ E\Big(a^2\Big)=\Big(a^2\Big)^2+2 a^2+1=a^4+2a^2+1\\\\ E\Big(-a^2\Big)=\Big(-a^2\Big)^2-2a^2+1=a^4-2a^2+1\\\\ E\Big(a^2\Big)-E\Big(-a^2\Big)=16 \Rightarrow a^4+2a^2+1-\Big(a^4-2a^2+1\Big)=16 \Rightarrow \\\\ \Rightarrow a^4+2a^2+1-a^4+2a^2-1=16 \Rightarrow 4a^2=16 \Rightarrow a^2=\frac{16}{4} \Rightarrow \\\\ \Rightarrow a^2=4 \Rightarrow a=\pm \sqrt{4} \Rightarrow a=\pm2$