Whakaoti i te 8(-0.5y+1)3y=31

$$8 ( - 0.5 y + 1 ) 3 y = 31$$

Whakaoti mō y

$$y=\frac{i\sqrt{57}}{6}+1\approx 1+1.258305739i$$

$$y=-\frac{i\sqrt{57}}{6}+1\approx 1-1.258305739i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 8 ki te 3, ka 24.

$$24\left(-0.5y+1\right)y=31$$

Whakamahia te āhuatanga tohatoha hei whakarea te 24 ki te -0.5y+1.

$$\left(-12y+24\right)y=31$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12y+24 ki te y.

$$-12y^{2}+24y=31$$

Tangohia te 31 mai i ngā taha e rua.

$$-12y^{2}+24y-31=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -12 mō a, 24 mō b, me -31 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-24±\sqrt{24^{2}-4\left(-12\right)\left(-31\right)}}{2\left(-12\right)}$$

Pūrua 24.

$$y=\frac{-24±\sqrt{576-4\left(-12\right)\left(-31\right)}}{2\left(-12\right)}$$

Whakareatia -4 ki te -12.

$$y=\frac{-24±\sqrt{576+48\left(-31\right)}}{2\left(-12\right)}$$

Whakareatia 48 ki te -31.

$$y=\frac{-24±\sqrt{576-1488}}{2\left(-12\right)}$$

Tāpiri 576 ki te -1488.

$$y=\frac{-24±\sqrt{-912}}{2\left(-12\right)}$$

Tuhia te pūtakerua o te -912.

$$y=\frac{-24±4\sqrt{57}i}{2\left(-12\right)}$$

Whakareatia 2 ki te -12.

$$y=\frac{-24±4\sqrt{57}i}{-24}$$

Nā, me whakaoti te whārite y=\frac{-24±4\sqrt{57}i}{-24} ina he tāpiri te ±. Tāpiri -24 ki te 4i\sqrt{57}.

$$y=\frac{-24+4\sqrt{57}i}{-24}$$

Whakawehe -24+4i\sqrt{57} ki te -24.

$$y=-\frac{\sqrt{57}i}{6}+1$$

Nā, me whakaoti te whārite y=\frac{-24±4\sqrt{57}i}{-24} ina he tango te ±. Tango 4i\sqrt{57} mai i -24.

$$y=\frac{-4\sqrt{57}i-24}{-24}$$

Whakawehe -24-4i\sqrt{57} ki te -24.

$$y=\frac{\sqrt{57}i}{6}+1$$

Kua oti te whārite te whakatau.

$$y=-\frac{\sqrt{57}i}{6}+1 y=\frac{\sqrt{57}i}{6}+1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 8 ki te 3, ka 24.

$$24\left(-0.5y+1\right)y=31$$

Whakamahia te āhuatanga tohatoha hei whakarea te 24 ki te -0.5y+1.

$$\left(-12y+24\right)y=31$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12y+24 ki te y.

$$-12y^{2}+24y=31$$

Whakawehea ngā taha e rua ki te -12.

$$\frac{-12y^{2}+24y}{-12}=\frac{31}{-12}$$

Mā te whakawehe ki te -12 ka wetekia te whakareanga ki te -12.

$$y^{2}+\frac{24}{-12}y=\frac{31}{-12}$$

Whakawehe 24 ki te -12.

$$y^{2}-2y=\frac{31}{-12}$$

Whakawehe 31 ki te -12.

$$y^{2}-2y=-\frac{31}{12}$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-2y+1=-\frac{31}{12}+1$$

Tāpiri -\frac{31}{12} ki te 1.

$$y^{2}-2y+1=-\frac{19}{12}$$

Tauwehea y^{2}-2y+1. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-1\right)^{2}=-\frac{19}{12}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(y-1\right)^{2}}=\sqrt{-\frac{19}{12}}$$

Whakarūnātia.

$$y-1=\frac{\sqrt{57}i}{6} y-1=-\frac{\sqrt{57}i}{6}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$y=\frac{\sqrt{57}i}{6}+1 y=-\frac{\sqrt{57}i}{6}+1$$