Question

Sa se determine x∈R astfel incat urmatoarele expresii sa fie definite:

1) √x²-3x (toata ecuatia este sub radical)
2) √2-4x (toata ec sub radical)
3) √-x2+3x-2 (radical de ordin 4 din *ce se gaseste sub radical*)
4) √3+x + √2x²-3x+1 (ambele ec sunt in intregime sub radical)
5) √x²-x+1 (toata ec e sub radical)

Multumesc anticipat!

Answer 1

1.  x² -3x >=0 <=> x(x-3) >=0 <=> x >=0 si x-3 >=0 => x >=3 dar x∈R => x∈[3,∞) sau x <=0 si x-3 <=0 => x <=3 dar x∈R => x∈(-∞,0] .

Asadar x∈(-∞,0] ∪ [3,∞) .

2.  2-4x >=0 <=> 4x <=2 <> x <=1/2 dar x∈R => x∈(-∞,1/2] .

3.  -x² +3x -2 >=0 => -x² +3x -2=0 => Δ=b²-4ac=9-4·(-1)·(-2)=9-8=1 .

x=-b+-√Δ /2a => x=-3-1 /-2 =2 sau x=-3+1 /-2 =1 => x∈[1,2] .

4.  2x² -3x +1 >=0 <=> 2x² -3x +1 =0 => Δ=b² -4ac=9-4·2·1=1 .

x=-b+-√Δ /2a => x=3-1 /4 =1/2 sau x=3+1 /4 =4/4=1 =>x∈[1/2,1] .

3+x >=0 => x >-3 <=> x∈[1/2,1] .

5.  x² -x +1 >=0 => x∈(-∞,∞) .