Whakaoti i te (14-x)(6x-24)/10=126

$$\frac{ (14-x)(6x-24) }{ 10 } =126$$

Whakaoti mō x (complex solution)

$$x=9+\sqrt{185}i\approx 9+13.601470509i$$

$$x=-\sqrt{185}i+9\approx 9-13.601470509i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua ki te 10.

$$\left(14-x\right)\left(6x-24\right)=126\times 10$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 14-x ki te 6x-24 ka whakakotahi i ngā kupu rite.

$$108x-336-6x^{2}=126\times 10$$

Whakareatia te 126 ki te 10, ka 1260.

$$108x-336-6x^{2}=1260$$

Tangohia te 1260 mai i ngā taha e rua.

$$108x-336-6x^{2}-1260=0$$

Tangohia te 1260 i te -336, ka -1596.

$$108x-1596-6x^{2}=0$$

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.

$$-6x^{2}+108x-1596=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, 108 mō b, me -1596 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-108±\sqrt{108^{2}-4\left(-6\right)\left(-1596\right)}}{2\left(-6\right)}$$

Pūrua 108.

$$x=\frac{-108±\sqrt{11664-4\left(-6\right)\left(-1596\right)}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-108±\sqrt{11664+24\left(-1596\right)}}{2\left(-6\right)}$$

Whakareatia 24 ki te -1596.

$$x=\frac{-108±\sqrt{11664-38304}}{2\left(-6\right)}$$

Tāpiri 11664 ki te -38304.

$$x=\frac{-108±\sqrt{-26640}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te -26640.

$$x=\frac{-108±12\sqrt{185}i}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{-108±12\sqrt{185}i}{-12}$$

Nā, me whakaoti te whārite x=\frac{-108±12\sqrt{185}i}{-12} ina he tāpiri te ±. Tāpiri -108 ki te 12i\sqrt{185}.

$$x=\frac{-108+12\sqrt{185}i}{-12}$$

Whakawehe -108+12i\sqrt{185} ki te -12.

$$x=-\sqrt{185}i+9$$

Nā, me whakaoti te whārite x=\frac{-108±12\sqrt{185}i}{-12} ina he tango te ±. Tango 12i\sqrt{185} mai i -108.

$$x=\frac{-12\sqrt{185}i-108}{-12}$$

Whakawehe -108-12i\sqrt{185} ki te -12.

$$x=9+\sqrt{185}i$$

Kua oti te whārite te whakatau.

$$x=-\sqrt{185}i+9 x=9+\sqrt{185}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua ki te 10.

$$\left(14-x\right)\left(6x-24\right)=126\times 10$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 14-x ki te 6x-24 ka whakakotahi i ngā kupu rite.

$$108x-336-6x^{2}=126\times 10$$

Whakareatia te 126 ki te 10, ka 1260.

$$108x-336-6x^{2}=1260$$

Me tāpiri te 336 ki ngā taha e rua.

$$108x-6x^{2}=1260+336$$

Tāpirihia te 1260 ki te 336, ka 1596.

$$108x-6x^{2}=1596$$

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.

$$-6x^{2}+108x=1596$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}+108x}{-6}=\frac{1596}{-6}$$

Mā te whakawehe ki te -6 ka wetekia te whakareanga ki te -6.

$$x^{2}+\frac{108}{-6}x=\frac{1596}{-6}$$

Whakawehe 108 ki te -6.

$$x^{2}-18x=\frac{1596}{-6}$$

Whakawehe 1596 ki te -6.

$$x^{2}-18x=-266$$

Whakawehea te -18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -9. Nā, tāpiria te pūrua o te -9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-18x+\left(-9\right)^{2}=-266+\left(-9\right)^{2}$$

Pūrua -9.

$$x^{2}-18x+81=-266+81$$

Tāpiri -266 ki te 81.

$$x^{2}-18x+81=-185$$

Tauwehea x^{2}-18x+81. 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}.

$$\left(x-9\right)^{2}=-185$$

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

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{-185}$$

Whakarūnātia.

$$x-9=\sqrt{185}i x-9=-\sqrt{185}i$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x=9+\sqrt{185}i x=-\sqrt{185}i+9$$