Question

1.Determinati ca x e R astfel incat 69 - 5(x -2) la a doua = 24.
2.Aratati ca a=-1 este solutie a ecuatiei 1 + x la a doua + (1+x) la a doua = -2x.
3.Rezolvati ecuatia : x la a doua = 1 + 3 + 5 + ... + 101.
4. Determinati x e R pentru care x-2 supra 2x + 3 + x +2 supra 2x -3=4
5. Rezolvati ecuatia 6(x -1) la a doua = (2+√10)• radical lung din 14 - 2√40.
Va rog repede dau maximul de puncte si coronita.

Answer 1

69 - 5(x -2)² = 24.
5(x -2)²=69-24
5(x -2)²=45
(x -2)²=9
x -2=9
x =11
2. 1 + x²+ (1+x)² = -2x
1 + x²+2x +(1+x)² =0
(x+1)²+(1+x)² =0
2(x+1)² =0
x+1 =0
x=-1

3. x² = 1 + 3 + 5 + ... + 101=51²      x=51
1+3+5+...2k-1=k²
101=2k-1
k=102/2=51


4.  x e R
(x-2)/(2x + 3) + (x +2)/(2x -3)=4
(x-2)((2x - 3) + (x +2)(2x +3)=4(2x-3)(2x+3)
2x²-4x - 3x+6 + 2x +4x +3x+6=4(4x²-9)
4x²+12=4(4x²-9)
x²+3=4x²-9
3x²=12
x²=4
x=2

5. 6(x -1)² = (2+√10)• √(14 - 2√40)√(14 - 2√40)=√(14 - √160)=√(14 + √196-160)/2-√(14 - √196-160)/2=
=√(14 + √36)/2-√(14 - √36)/2=√(14 + 6)/2-√(14 - 6)/2=√10-√4=√10-2
6(x -1)² = (2+√10)• (√10-2)
6(x -1)² = (√10+2)• (√10-2)
6(x -1)² = 10-4
6(x -1)² = 6
(x -1)² =1
x -1 =1
x=2