Question

Rezolvati in R ecuatiile:
a) l 3x-7 l=8
b) l x l*l x-3 l=4
c) l 2x la a 2 +17x+658 l= -2rdical din 19
d) l 2x(x-0.5) l=3

Answer 1

a)

[tex]\it |3x-7| =8\Leftrightarrow 3x-7=\pm8 \Leftrightarrow3x-7\in\{-8,\ 8\}|_{+7} \Leftrightarrow \\\;\\ \Leftrightarrow3x\in\{-1,\ 15\}|_{:3} \Leftrightarrow x\in \left\{-\dfrac{1}{3},\ 5 \right\}[/tex]

b)

[tex]\it |x||x-3 |=4 \Leftrightarrow |x(x-3)| =4 \Leftrightarrow x(x-3) = \pm4 \Leftrightarrow x^2-3x = \pm4 \\\;\\ I) x^2-3x = -4 \Leftrightarrow x^2-3x+4=0|_{\cdot4} \Leftrightarrow 4x^2-12x+16=0 \Leftrightarrow \\\;\\ \Leftrightarrow 4x^2-12x+9+7=0 \Leftrightarrow (2x-3)^2+7=0\ (Fals)[/tex]

(2x-3)² +7 ≥ 7, pentru orice x - real


[tex]II) \it x^2-3x=4 \Leftrightarrow x^2-3x-4=0 \Leftrightarrow x^2+x-4x-4=0 \Leftrightarrow \\\;\\ \Leftrightarrow x(x+1) -4(x+1) =0 \Leftrightarrow (x+1)(x-4)=0 \Leftrightarrow \\\;\\ \Leftrightarrow \begin{cases} \it x+1=0 \Longrightarrow x = -1 \\\;\\ \it x-4=0 \Longrightarrow x = 4\end{cases}[/tex]



c) |2x² +17x + 658|= -2√19  ⇒ S = ∅, deoarece modulul nu poate fi negativ.