Question

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Answer 1

Răspuns

poza

Explicație pas cu pas.

Answer 2

$\it a)\ E(x)=4x^2-1=2^2x^2-1=(2x-1)(2x+1)\\ \\ F(x)=2x^2-x=x(2x-1)\\ \\ \\ E\left(\dfrac{1}{2}\right)=(2\cdot\dfrac{1}{2}-1)(2\cdot\dfrac{1}{2}+1)=(1-1)(1+1)=0\cdot2=0\\ \\ \\ F\left(\dfrac{1}{2}\right)=\dfrac{1}{2}(2\cdot\dfrac{1}{2}-1)=\dfrac{1}{2}(1-1)=\dfrac{1}{2}\cdot0=0\\ \\ \\ Deci,\ E\left(\dfrac{1}{2}\right)=F\left(\dfrac{1}{2}\right)=0$

$\it b)\ E(4)\cdot F(4)=(4\cdot4^2-1)\cdot(2\cdot4^2-4)=63\cdot28=9\cdot7\cdot7\cdot4=3^2\cdot7^2\cdot2^2=\\ \\ =(3\cdot7\cdot2)^2=42^2\ (p\breve{a}trat\ perfect)$

$\it c)\ F(x)=2x^2-x=x(2x-1)\\ \\ d)\ E(x)-F(x)=(2x-1)(2x+1)-x(2x-1)=(2x-1)(2x+1-x)=\\ \\ =(2x-1)(x+1)\ \ \ (*)$

$\it e)\ (*) \Rightarrow E(n)-F(n)=(2n-1)(n+1)=0 \Longrightarrow\begin{cases}\it 2n-1=0\Rightarrow n=\dfrac{1}{2}\notin\mathbb{Z}\\ \\ n+1=0\Rightarrow n=-1\in\mathbb{Z}\end{cases}$

Deci, E(n) - F(n) = 0 pentru n = -1.