Question

Simplificati raportul 3 ori x(la puterea 2) + 5x -2 totul supra 3 ori x (la puterea 2) -7x + 2

Answer 1

La numarator: 3x^2 + 5x - 2 = 3x^2+6x-x-2 = 3x(x+2)-(x+2) = (3x-1)(x+2)

La numitor : 3x^2 - 7x + 2 = 3x^2-6x-x+2 = 3x(x-2)-(x-2) = (3x-1)(x-2)

Se simplifica cu 3x - 1 si rezulta (x + 2)(x - 2)

Answer 2

$\displaystyle\\ \frac{3x^2+5x-2}{3x^2-7x+2}=?\\\\ \text{Descompunem in factori cele 2 polinoame.}\\\\ 3x^2+5x-2=0\\\\ x_{12}=\frac{-5\pm \sqrt{5^2 +4\cdot3\cdot2}}{2\cdot 3}=\\\\ =\frac{-5\pm \sqrt{25 +24}}{6}= \frac{-5\pm \sqrt{49}}{6}=\frac{-5\pm 7}{6}\\\\ x_1 = \frac{-5-7}{6}=\frac{-12}{6}=\boxed{-2}\\\\ x_2 = \frac{-5+7}{6}=\frac{2}{6}=\boxed{\frac{1}{3}}\\\\ \Longrightarrow~~~3x^2+5x-2=\Big(x+2\Big)\Big(3x-1\Big)\\\\ -------------$

$\displaystyle\\ 3x^2-7x+2=0\\\\ x_{34}=\frac{7\pm\sqrt{7^2-4\cdot3\cdot2}}{2\cdot3}=\\\\ =\frac{7\pm\sqrt{49-24}}{6}= \frac{7\pm\sqrt{49-24}}{6}=\frac{7\pm\sqrt{25}}{6}= \frac{7\pm5}{6}\\\\ x_3=\frac{7+5}{6}=\frac{12}{6}=\boxed{2}\\\\ x_4=\frac{7-5}{6}=\frac{2}{6}=\boxed{\frac{1}{3}}\\\\ \Longrightarrow~~~3x^2-7x+2=\Big(x-2\Big)\Big(3x-1\Big)\\\\ -------------------$

$\displaystyle\\ \Longrightarrow~~~\frac{3x^2+5x-2}{3x^2-7x+2}= \frac{\Big(x+2\Big)\Big(3x-1\Big)}{\Big(x-2\Big)\Big(3x-1\Big)}= \frac{\Big(x+2\Big)}{\Big(x-2\Big)}=\boxed{\frac{x+2}{x-2}}$