Whakaoti i te 6/x^2-12/x=3

$$\frac { 6 } { x ^ { 2 } } - \frac { 12 } { x } = 3$$

Whakaoti mō x (complex solution)

$$x=\sqrt{6}-2\approx 0.449489743$$

$$x=-\left(\sqrt{6}+2\right)\approx -4.449489743$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2},x.

$$6-x\times 12=3x^{2}$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$6-x\times 12-3x^{2}=0$$

Whakareatia te -1 ki te 12, ka -12.

$$6-12x-3x^{2}=0$$

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.

$$-3x^{2}-12x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, -12 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-12\right)±\sqrt{144+12\times 6}}{2\left(-3\right)}$$

Whakareatia 12 ki te 6.

$$x=\frac{-\left(-12\right)±\sqrt{144+72}}{2\left(-3\right)}$$

Tāpiri 144 ki te 72.

$$x=\frac{-\left(-12\right)±\sqrt{216}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 216.

$$x=\frac{-\left(-12\right)±6\sqrt{6}}{2\left(-3\right)}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±6\sqrt{6}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{12±6\sqrt{6}}{-6}$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{6}}{-6} ina he tāpiri te ±. Tāpiri 12 ki te 6\sqrt{6}.

$$x=\frac{6\sqrt{6}+12}{-6}$$

Whakawehe 12+6\sqrt{6} ki te -6.

$$x=-\left(\sqrt{6}+2\right)$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{6}}{-6} ina he tango te ±. Tango 6\sqrt{6} mai i 12.

$$x=\frac{12-6\sqrt{6}}{-6}$$

Whakawehe 12-6\sqrt{6} ki te -6.

$$x=\sqrt{6}-2$$

Kua oti te whārite te whakatau.

$$x=-\left(\sqrt{6}+2\right) x=\sqrt{6}-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2},x.

$$6-x\times 12=3x^{2}$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$6-x\times 12-3x^{2}=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-x\times 12-3x^{2}=-6$$

Whakareatia te -1 ki te 12, ka -12.

$$-12x-3x^{2}=-6$$

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.

$$-3x^{2}-12x=-6$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}-12x}{-3}=-\frac{6}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\left(-\frac{12}{-3}\right)x=-\frac{6}{-3}$$

Whakawehe -12 ki te -3.

$$x^{2}+4x=-\frac{6}{-3}$$

Whakawehe -6 ki te -3.

$$x^{2}+4x=2$$

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

$$x^{2}+4x+2^{2}=2+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=2+4$$

Tāpiri 2 ki te 4.

$$x^{2}+4x+4=6$$

Tauwehea x^{2}+4x+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}.

$$\left(x+2\right)^{2}=6$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{6}$$

Whakarūnātia.

$$x+2=\sqrt{6} x+2=-\sqrt{6}$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$

Whakaoti mō x

$$x=\sqrt{6}-2\approx 0.449489743$$

$$x=-\sqrt{6}-2\approx -4.449489743$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2},x.

$$6-x\times 12=3x^{2}$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$6-x\times 12-3x^{2}=0$$

Whakareatia te -1 ki te 12, ka -12.

$$6-12x-3x^{2}=0$$

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.

$$-3x^{2}-12x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, -12 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-12\right)±\sqrt{144+12\times 6}}{2\left(-3\right)}$$

Whakareatia 12 ki te 6.

$$x=\frac{-\left(-12\right)±\sqrt{144+72}}{2\left(-3\right)}$$

Tāpiri 144 ki te 72.

$$x=\frac{-\left(-12\right)±\sqrt{216}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 216.

$$x=\frac{-\left(-12\right)±6\sqrt{6}}{2\left(-3\right)}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±6\sqrt{6}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{12±6\sqrt{6}}{-6}$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{6}}{-6} ina he tāpiri te ±. Tāpiri 12 ki te 6\sqrt{6}.

$$x=\frac{6\sqrt{6}+12}{-6}$$

Whakawehe 12+6\sqrt{6} ki te -6.

$$x=-\left(\sqrt{6}+2\right)$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{6}}{-6} ina he tango te ±. Tango 6\sqrt{6} mai i 12.

$$x=\frac{12-6\sqrt{6}}{-6}$$

Whakawehe 12-6\sqrt{6} ki te -6.

$$x=\sqrt{6}-2$$

Kua oti te whārite te whakatau.

$$x=-\left(\sqrt{6}+2\right) x=\sqrt{6}-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2},x.

$$6-x\times 12=3x^{2}$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$6-x\times 12-3x^{2}=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-x\times 12-3x^{2}=-6$$

Whakareatia te -1 ki te 12, ka -12.

$$-12x-3x^{2}=-6$$

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.

$$-3x^{2}-12x=-6$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}-12x}{-3}=-\frac{6}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\left(-\frac{12}{-3}\right)x=-\frac{6}{-3}$$

Whakawehe -12 ki te -3.

$$x^{2}+4x=-\frac{6}{-3}$$

Whakawehe -6 ki te -3.

$$x^{2}+4x=2$$

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

$$x^{2}+4x+2^{2}=2+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=2+4$$

Tāpiri 2 ki te 4.

$$x^{2}+4x+4=6$$

Tauwehea x^{2}+4x+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}.

$$\left(x+2\right)^{2}=6$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{6}$$

Whakarūnātia.

$$x+2=\sqrt{6} x+2=-\sqrt{6}$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$