1. 3(x-y)-(2x+y)+(y-x)=3x-3y-2x-y+y-x=-3y.
2. [tex]( 3x^{2} y)* (\frac{1}{3} xy^{4}) :( x^{2} y^{2} )=( x^{3} y^{5} ):( x^{2} y^{5})= x.[/tex]
3. [tex](x-1)(x+3)= x^{2} +3x-x-3= x^{2} +2x+3.[/tex]
4. Fie a, (a+1) si (a+2) numerele.
[tex] a^{2} + (a+1)^{2}+ (a+2)^{2} = a^{2}+ a^{2}+2a+1+ a^{2}+4a+4= \\ \\ =3 a^{2} +6a+5= 3a^{2}+6a+3+2=3( a^{2}+2a+1)+2= M_{3} +2[/tex]
=> Suma patratelor a 3 numere intregi consecutive da restul 2 la impartirea cu 3.
5.
[tex](1+ \sqrt{2})(1- \sqrt{2})+(3+ \sqrt{2}) ^{2} -6 \sqrt{2} = \\ =1-2+9+6 \sqrt{2}+2- 6\sqrt{2}= \\ =10. [/tex]
6.
[tex](x+ \frac{1}{x}) ^{2} = 3^{2}<=> \\ <=> x^{2} +2*x* \frac{1}{x} + \frac{1}{ x^{2} }=9<=> \\ <=> x^{2} +2+ \frac{1}{ x^{2} } =9=> x^{2} + \frac{1}{ x^{2} } =7.[/tex]
7.
[tex]a)16-8x+ x^{2} =(4-x) ^{2} \\ b) x^{3} -25x=x( x^{2} -25)=x(x-5)(x+5) \\ c) x^{2} -2xy+ y^{2}- z^{2}= (x-y)^{2} - z^{2}=(x-y-z)(x-y+z) \\ d)ax-ay-bx+by=a(x-y)-b(x-y)=(x-y)(a-b) [/tex]
8.
[tex]4 x^{2} =100=> x^{2} =25=> x^{2} -25=0<=>(x-5)(x+5)=0=> \\ \\ =>x=5~sau~x=-5.[/tex]
Solutie: x∈{-5 ; 5}.
