Question

Va rog frumos 2,3,4.

Answer 1

$2.\ a=\frac{1}{\sqrt{5}+\sqrt{3}}-(\frac{2}{\sqrt{5}-\sqrt{3}})^{-1}\\\\Rationalizam\ prima\ fractie\ amplificam\ cu\ conjugatul,\ iar\ pe\ a\ doua\ o\ intoarcem,\ caci\ \frac{a}{b})^{-1}=\frac{b}{a}$

$a=\frac{\sqrt{5}-\sqrt{3}}{(\sqrt{5}-\sqrt{3})*(\sqrt{5}+\sqrt{3})}-\frac{\sqrt{5}-\sqrt{3}}{2}\\\\a=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}^{2}-\sqrt{3}^{2}}-\frac{\sqrt{5}-\sqrt{3}}{2}\\\\a=\frac{\sqrt{5}-\sqrt{3}}{5-3}-\frac{\sqrt{5}-\sqrt{3}}{2}\\\\a=\frac{\sqrt{5}-\sqrt{3}}{2}-\frac{\sqrt{5}-\sqrt{3}}{2}\\\\a=\frac{\sqrt{5}-\sqrt{3}-\sqrt{5}+\sqrt{3}}{2}\\\\a=\frac{0}{2}=0=>\ a\ este\ rational$

3. A={x∈R/ |x+4| ≤ 5}

|x+4| ≤5 → -5 ≤ x+4 ≤ 5, scadem 4

-5-4 ≤ x ≤ 5-4

-9 ≤ x ≤ 1 → A=[-9,1]

4. a) x³-2x²-4x+8= x²×(x-2) -4×(x-2)= (x²-1)×(x-2)=(x+1)×(x-1)×(x-2)

b) (x-2)²×(x+2)-x-2 = (x-2)²×(x+2) - (x+2) = (x+2)×[(x-2)²-1] = (x+2)×(x-2-1)(x-2+1) = (x+2)×(x-3)×(x-1) (Ceea ce trebuia demonstrat)

Observatii: M-am folosit de formula (a-b)×(a+b)=a² - b²/ a² - b²=(a-b)×(a+b).