Respuesta :

[tex]\bf y^2-2y=-\cfrac{9}{2}\implies y^2-2y+\cfrac{9}{2}=0\\\\ -------------------------------\\\\ \qquad \qquad \textit{quadratic formula}\\\\ \begin{array}{lccclll} &{{ 1}}x^2&{{ -2}}x&{{ +\cfrac{9}{2}}}&=0\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array} \qquad \qquad y= \cfrac{ - {{ b}} \pm \sqrt { {{ b}}^2 -4{{ a}}{{ c}}}}{2{{ a}}}[/tex]

[tex]\bf y=\cfrac{-(-2)\pm\sqrt{(-2)^2-4(1)\left( \frac{9}{2} \right)}}{2(1)}\implies y=\cfrac{2\pm\sqrt{4-18}}{2} \\\\\\ y=1\pm\sqrt{-14}\implies y=1\pm\sqrt{-1\cdot 14}\implies y=1\pm\sqrt{-1}\cdot \sqrt{14} \\\\\\ y=1\pm i\sqrt{14}[/tex]

Answer:

[tex]y= \frac{2+\isqrt{14}}{2}[/tex]

[tex]y= \frac{2-\isqrt{14}}{2}[/tex]

Step-by-step explanation:

[tex]y^2 - 2y = \frac{-9}{2}[/tex]

add 9/2 on both sides

[tex]y^2 - 2y + \frac{9}{2}=0[/tex]

Apply quadratic formula and solve for y

a= 1, b=-2 and c= 9/2

[tex]y= \frac{-b+-\sqrt{b^2-4ac}}{2a}[/tex]

[tex]y= \frac{2+-\sqrt{(-2)^2-4*1*9/2}}{2(1)}[/tex]

[tex]y= \frac{2+-i\sqrt{14}}{2}[/tex]

WE cannot simplify it further

[tex]y= \frac{2+i\sqrt{14}}{2}[/tex]

[tex]y= \frac{2-i\sqrt{14}}{2}[/tex]