Whakaoti i te 5/3+(24-8x)/12x=2/3

$$5 / 3 + ( 24 - 8 x ) / 12 x = 2 / 3$$

Whakaoti mō x

$$x = \frac{\sqrt{15} + 3}{2} \approx 3.436491673$$

$$x=\frac{3-\sqrt{15}}{2}\approx -0.436491673$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,12.

$$20+\left(24-8x\right)x=8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 24-8x ki te x.

$$20+24x-8x^{2}=8$$

Tangohia te 8 mai i ngā taha e rua.

$$20+24x-8x^{2}-8=0$$

Tangohia te 8 i te 20, ka 12.

$$12+24x-8x^{2}=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-8x^{2}+24x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, 24 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-8\right)\times 12}}{2\left(-8\right)}$$

Pūrua 24.

$$x=\frac{-24±\sqrt{576-4\left(-8\right)\times 12}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-24±\sqrt{576+32\times 12}}{2\left(-8\right)}$$

Whakareatia 32 ki te 12.

$$x=\frac{-24±\sqrt{576+384}}{2\left(-8\right)}$$

Tāpiri 576 ki te 384.

$$x=\frac{-24±\sqrt{960}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 960.

$$x=\frac{-24±8\sqrt{15}}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{-24±8\sqrt{15}}{-16}$$

Nā, me whakaoti te whārite x=\frac{-24±8\sqrt{15}}{-16} ina he tāpiri te ±. Tāpiri -24 ki te 8\sqrt{15}.

$$x=\frac{8\sqrt{15}-24}{-16}$$

Whakawehe -24+8\sqrt{15} ki te -16.

$$x=\frac{3-\sqrt{15}}{2}$$

Nā, me whakaoti te whārite x=\frac{-24±8\sqrt{15}}{-16} ina he tango te ±. Tango 8\sqrt{15} mai i -24.

$$x=\frac{-8\sqrt{15}-24}{-16}$$

Whakawehe -24-8\sqrt{15} ki te -16.

$$x=\frac{\sqrt{15}+3}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{3-\sqrt{15}}{2} x=\frac{\sqrt{15}+3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,12.

$$20+\left(24-8x\right)x=8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 24-8x ki te x.

$$20+24x-8x^{2}=8$$

Tangohia te 20 mai i ngā taha e rua.

$$24x-8x^{2}=8-20$$

Tangohia te 20 i te 8, ka -12.

$$24x-8x^{2}=-12$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-8x^{2}+24x=-12$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8x^{2}+24x}{-8}=-\frac{12}{-8}$$

Mā te whakawehe ki te -8 ka wetekia te whakareanga ki te -8.

$$x^{2}+\frac{24}{-8}x=-\frac{12}{-8}$$

Whakawehe 24 ki te -8.

$$x^{2}-3x=-\frac{12}{-8}$$

Whakahekea te hautanga \frac{-12}{-8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-3x=\frac{3}{2}$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=\frac{3}{2}+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=\frac{3}{2}+\frac{9}{4}$$

Tāpiri \frac{3}{2} ki te \frac{9}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-3x+\frac{9}{4}=\frac{15}{4}$$

Tauwehea x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{15}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{15}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{\sqrt{15}}{2} x-\frac{3}{2}=-\frac{\sqrt{15}}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{15}+3}{2} x=\frac{3-\sqrt{15}}{2}$$