Whakaoti i te (12-x)(20+x)=1750

$$(12-x)(20+x)=1750$$

Whakaoti mō x (complex solution)

$$x=-3\sqrt{166}i-4\approx -4-38.65229618i$$

$$x=-4+3\sqrt{166}i\approx -4+38.65229618i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 12-x ki te 20+x ka whakakotahi i ngā kupu rite.

$$240-8x-x^{2}=1750$$

Tangohia te 1750 mai i ngā taha e rua.

$$240-8x-x^{2}-1750=0$$

Tangohia te 1750 i te 240, ka -1510.

$$-1510-8x-x^{2}=0$$

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.

$$-x^{2}-8x-1510=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -8 mō b, me -1510 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\left(-1510\right)}}{2\left(-1\right)}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\left(-1510\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\left(-1510\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -1510.

$$x=\frac{-\left(-8\right)±\sqrt{64-6040}}{2\left(-1\right)}$$

Tāpiri 64 ki te -6040.

$$x=\frac{-\left(-8\right)±\sqrt{-5976}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -5976.

$$x=\frac{-\left(-8\right)±6\sqrt{166}i}{2\left(-1\right)}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±6\sqrt{166}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{8±6\sqrt{166}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{8±6\sqrt{166}i}{-2} ina he tāpiri te ±. Tāpiri 8 ki te 6i\sqrt{166}.

$$x=\frac{8+6\sqrt{166}i}{-2}$$

Whakawehe 8+6i\sqrt{166} ki te -2.

$$x=-3\sqrt{166}i-4$$

Nā, me whakaoti te whārite x=\frac{8±6\sqrt{166}i}{-2} ina he tango te ±. Tango 6i\sqrt{166} mai i 8.

$$x=\frac{-6\sqrt{166}i+8}{-2}$$

Whakawehe 8-6i\sqrt{166} ki te -2.

$$x=-4+3\sqrt{166}i$$

Kua oti te whārite te whakatau.

$$x=-3\sqrt{166}i-4 x=-4+3\sqrt{166}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 12-x ki te 20+x ka whakakotahi i ngā kupu rite.

$$240-8x-x^{2}=1750$$

Tangohia te 240 mai i ngā taha e rua.

$$-8x-x^{2}=1750-240$$

Tangohia te 240 i te 1750, ka 1510.

$$-8x-x^{2}=1510$$

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.

$$-x^{2}-8x=1510$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-8x}{-1}=\frac{1510}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=\frac{1510}{-1}$$

Whakawehe -8 ki te -1.

$$x^{2}+8x=\frac{1510}{-1}$$

Whakawehe 1510 ki te -1.

$$x^{2}+8x=-1510$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+8x+4^{2}=-1510+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=-1510+16$$

Tāpiri -1510 ki te 16.

$$x^{2}+8x+16=-1494$$

Tauwehea x^{2}+8x+16. 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}.

$$\left(x+4\right)^{2}=-1494$$

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

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{-1494}$$

Whakarūnātia.

$$x+4=3\sqrt{166}i x+4=-3\sqrt{166}i$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$x=-4+3\sqrt{166}i x=-3\sqrt{166}i-4$$