Question

Sa se determine multimea A={x apartine Z | √4x+7/x+4 apartine N } gen tot calcul ala e sub radical

Answer 1

Explicație pas cu pas:

$\sqrt{ \dfrac{4x + 7}{x + 4} } = \sqrt{ \dfrac{4x + 16 - 16 + 7}{x + 4} } = \\ = \sqrt{ \dfrac{4(x + 4) - 9}{x + 4} } = \sqrt{4 - \dfrac{9}{x + 4} } \in \mathbb{N}$

$\dfrac{9}{x + 4} \in \mathbb{Z} \iff (x + 4) \in \mathcal{M_{9}}$

$\iff (x + 4) \in \Big\{ - 9; - 3; - 1; 1; 3; 9\Big\} \\ \implies x \in \Big\{ - 13; - 7; - 5; - 3; - 1; 5\Big\}$

se verifică:

$\begin{cases} x = - 13 \iff \sqrt{4 + 1} = \sqrt{5} \not \in \mathbb{N}\\ x = - 7 \iff \sqrt{4 + 3} = \sqrt{7} \not \in \mathbb{N}\\ x = - 5 \iff \sqrt{4 + 9} = \sqrt{13} \not \in \mathbb{N}\\ x = - 3 \iff \sqrt{4 - 9} = \sqrt{ - 5} \not \in \mathbb{N}\\ x = - 1 \iff \sqrt{4 - 3} = \sqrt{1} = 1 \in \mathbb{N} \\ x = - 1 \iff \sqrt{4 - 1} = \sqrt{3} \not \in \mathbb{N} \end{cases}$

$\implies \bf A = \{-1\}$