Question

2xpatrat +6x +5 =y 7xpatrat -12x + 18 =y (sistem)

Answer 1

$\left\{ \begin{array}{c} 2x^2+6x+5 = y \\ 7x^2-12x+18 = y \end{array} \right |\Rightarrow 2x^2+6x+5 = 7x^2-12x+18 \Rightarrow \\ \\ \Rightarrow 0 = 7x^2-2x^2-12x-6x+18-5 \Rightarrow 5x^2-18x+13= 0 \\ \\ \Delta = (-18)^2 - 4\cdot5\cdot13 = 324 - 260 = 64 \Rightarrow x_{1,2} = \dfrac{-(-18)\pm \sqrt{64} }{2\cdot 5} \Rightarrow \\ \\ \Rightarrow x_{1,2} = \dfrac{18\pm8}{10} \Rightarrow x_1 = 1, \quad x_2 = \dfrac{13}{5}$

$\boxed{1} \quad x = 1 \Rightarrow 2\cdot1^2 + 6\cdot1+5 = y \Rightarrow y = 2+6+5 \Rightarrow y = 13 \Rightarrow \\\Rightarrow (x,y) = \Big(1, 13\Big) \\ \\ \boxed{2} \quad x = \dfrac{13}{5} \Rightarrow 2\cdot\Big(\dfrac{13}{5}\Big)^2 + 6\cdot \dfrac{13}{5}+5 = y \Rightarrow 2\cdot \dfrac{169}{25} + \dfrac{78}{5} +5 = y \Rightarrow$
$\Rightarrow 2\cdot 169 + 5\cdot 78+25\cdot 5 = 25y \Rightarrow 853 = 25y \Rightarrow y = \dfrac{853}{25} \Rightarrow \\ \Rightarrow (x,y) = \Big(\dfrac{13}{5}, \dfrac{853}{25}\Big)$

$\ Din \boxed{1} \cup \boxed{2} \Rightarrow \Big(x,y\Big) = \left\{ \Big(1, 13\Big) ;\Big(\dfrac{13}{5}, \dfrac{853}{25}\Big)\right\}$


Am atasat si o imagine in care se observa ca acestea sunt cele doua puncte de intersectie ale celor doua functii.