Răspuns :
Ridicam la patrat :
[tex]\left(\sqrt{2x+1}+\sqrt{x-3}\right)^2=\left(2\sqrt{x}\right)^2[/tex]
\\[tex]\left(\sqrt{2x+1}\right)^2+2\sqrt{2x+1}\sqrt{x-3}+\left(\sqrt{x-3}\right)^2=\left(2\sqrt{x}\right)^2\\[/tex]\\
[tex]2\sqrt{2x+1}\sqrt{x-3}+3x-2=4x[/tex]\\
[tex]2\sqrt{2x+1}\sqrt{x-3}=x+2[/tex]
Ridicam la patrat
[tex]\left(2\sqrt{2x+1}\sqrt{x-3}\right)^2=\left(x+2\right)^2[/tex]\\
[tex]8x^2-20x-12=x^2+4x+4[/tex]\\
[tex]7x^2-24x-16=0[/tex]
Rezolvam ecuatia si avem a=7
b=-24
c=-16
Δ=576-4*7+(-16) =576+448=1024
√Δ=32 ->x1=4
x2=-4/7
Verificam solutiile
Pt x1=4 avem 3+1 =2*2 (A)
Pt x2=-4/7 avem √-1/7 +.... (F)
In concluzie singura solutie este x=4
[tex]\left(\sqrt{2x+1}+\sqrt{x-3}\right)^2=\left(2\sqrt{x}\right)^2[/tex]
\\[tex]\left(\sqrt{2x+1}\right)^2+2\sqrt{2x+1}\sqrt{x-3}+\left(\sqrt{x-3}\right)^2=\left(2\sqrt{x}\right)^2\\[/tex]\\
[tex]2\sqrt{2x+1}\sqrt{x-3}+3x-2=4x[/tex]\\
[tex]2\sqrt{2x+1}\sqrt{x-3}=x+2[/tex]
Ridicam la patrat
[tex]\left(2\sqrt{2x+1}\sqrt{x-3}\right)^2=\left(x+2\right)^2[/tex]\\
[tex]8x^2-20x-12=x^2+4x+4[/tex]\\
[tex]7x^2-24x-16=0[/tex]
Rezolvam ecuatia si avem a=7
b=-24
c=-16
Δ=576-4*7+(-16) =576+448=1024
√Δ=32 ->x1=4
x2=-4/7
Verificam solutiile
Pt x1=4 avem 3+1 =2*2 (A)
Pt x2=-4/7 avem √-1/7 +.... (F)
In concluzie singura solutie este x=4
