Question

AJUTORRR !!!
1) Calculati media aritmetica a numerelor: 5√3-2√5 ; √8-2√15;
2) Ararati ca (2x+1)³-8x-4=(2x-1)(2x+1)(2x+3)
3) Aratati ca expresia E(x)= (x²+2x+2)(x²+2x+6)+7 este strict pozitiva pt orice x∈R. Determinati valoarea ecuatiile.
AJUTATI-MA DAU CORONITA :o3 :(

Answer 1

$1)~ m_{a}= \frac{(5 \sqrt{3}-2 \sqrt{5})+( \sqrt{8}-2 \sqrt{15}) }{2} = \frac{ 5\sqrt{3}- 2\sqrt{5}+2 \sqrt{2} -2 \sqrt{15} }{2} . \\ \\ 2)~(2x+1) ^{3} -8x-4=(2x+1) ^{3}-4(2x+1)= \\ =(2x+1)[(2 x+1) ^{2}-4 ]=(2x+1)(2x-1)(2x+3) \\ \\ 3)~Notam~ x^{2} +2x+2=a. \\ E(x)=a(a+4)+7= a^{2} +4a+7= a^{2} +4a+4+3=(a+2) ^{2}+3. \\ \\ Deci~E(x)\ \textgreater \ 0.$

Ca sa nu sterg si altceva din greseala, o sa scriu mai jos rezolvarea primul exercitiu:

$1)~ \sqrt{8-2 \sqrt{15} } = \sqrt{5-2 \sqrt{15}+3 }= \sqrt{( \sqrt{5}- \sqrt{3})^{2} } =| \sqrt{5}- \sqrt{3}|= \\ = \sqrt{5}- \sqrt{3} \\ \frac{2}{ \sqrt{5}+ \sqrt{3} }= \frac{2( \sqrt{5} - \sqrt{3}) }{5-3}= \frac{2( \sqrt{5}- \sqrt{3} )}{2}= \sqrt{5}- \sqrt{3} \\ \\ m_{a} = \frac{( 5 \sqrt{3}-2 \sqrt{5}) +(\sqrt{5}- \sqrt{3})+( \sqrt{5}- \sqrt{3}) }{3}= \frac{7 \sqrt{3} -4 \sqrt{5} }{3} .$