Question

rezolvati in R ecuatia:
(9-x²)√x2 - 2x - 15 = 0 (√x2 - 2x - 15) toate sunt sub radical.

Answer 1

$(9- x^{2} ) \sqrt{ x^{2} -2x-15}=0=\ \textgreater \ 9- x^{2} =0 ~SAU~ \sqrt{ x^{2} -2x-15}=0.$

$9- x^{2} =0\ \textless \ =\ \textgreater \ (3-x)(3+x)=0=\ \textgreater \ x=3~sau~x=-3.$

$\sqrt{ x^{2} -2x-15}=0\ \textless \ =\ \textgreater \ \sqrt{ x^{2} +3x-5x-15}=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ \sqrt{x(x+3)-5(x+3)}=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ \sqrt{(x+3)(x-5)}=0=\ \textgreater \ x=-3~sau~x=5.$

$\sqrt{ x^{2} -2x-15}$ este definit pentru $x^{2} -2x-15 \geq 0$.

La inceput am obtinut solutia x=3, dar trebuie sa verificam daca pentru aceasta radicalul este definit:

x=3 => $\sqrt{ x^{2} -2x-15}= \sqrt{ (x+3)(x-5)}= \sqrt{(3+3)(3-5)}= \sqrt{6*(-2)}= \\ = \sqrt{(-12)}$-nedefinit.

Deci x=3 nu convine. (Pentru ca atunci radicalul este nedefinit).

Solutie: x∈{-3 ; 5}.