Question

Rezolvati in R ecuatia a) (3x-1)^2-4(5x+6)=9(x^2-x+1)+2
b) |x|=2x+1
c) |2-x|=-0,5x+4

Answer 1

a)
(3x-1)^2-4(5x+6)=9(x^2-x+1)+2 <=> 9x^2-6x+1-(20x+24)=9x^2-9x+9+2 <=> 9x^2-6x+1-20x-24=9x^2-9x+11 <=> 9x^2-26x-23=9x^2-9x+11 <=> 17x+34=0 <=> 17x=-34 <=> x=-2.

b)
Daca x>=0: |x|=2x+1 <=> x=2x+1 <=> -x=1 <=> x=-1, ceea ce NU convine din cauza conditiei x>=0.
Daca x<0: |x|=2x+1 <=> -x=2x+1 <=> 3x=-1 <=> x=-1/3, ceea ce convine, deoarece -1/3<0.
=> solutia ecuatiei |x|=2x+1 este x=-1/3.

c)
Daca 2-x>=0 ( <=> x<=2): |2-x|=-0,5x+4 <=> 2-x=-0,5x+4 <=> 0,5x=-2 <=> x=-4, ceea ce convine deoarece -4<=2.
Daca 2-x<0 ( <=> x>2): |2-x|=-0,5x+4 <=> x-2=-0,5x+4 <=> 1,5x=6 <=> x=4, ceea ce convine deoarece 4>2.
=> solutiile ecuatiei |2-x|=-0,5x+4 sunt x=-4 si x=4.