Question

Rezolvati ecuatia |x-2y| + x²-4x+4=0

Answer 1

|x-2y|+x²-4x+4=0 ©
|x-2y|=4x-x²-4
1) x-2y=4x-x²-4
x²-3x+4-2y=0
∆1=9-4(4-2y)=9-16+8y=8y-7
3) x-2y=-(4x-x²-4)
x-2y=x²-4x+4
-x²+5x-2y-4=0
∆2=25+4(-2y-4)=25-8y-16=9-8y

Pentru ecuatia © , este necesar ca ∆1=∆2, atunci
8y-7=9-8y
16y=16
y=1
→ ecuatia © are forma
|x-2|+x²-4x+4=0
|x-2|=4x-x²-4
1) x-2=4x-x²-4
x²-3x+2=0
∆=9-8=1
x1,2=(3±1)/2
x1=1
x2=2
2) x-2=-(4x-x²-4)
x-2=x²-4x+4
-x²+5x-6=0
∆=25-24=1
x1,2=(-5±1)/-2
x1=3
x2=2

Observam ca solutia comuna a ambelor ecuatii este 2. Deci x=2
Solutia finala este:
S={(2;1)}