Whakaoti i te x(x-6sqrt{2})+65=0

$$x(x-6 \sqrt{ 2 } )+65=0$$

Whakaoti mō x (complex solution)

$$x=3\sqrt{2}+\sqrt{47}i\approx 4.242640687+6.8556546i$$

$$x=-\sqrt{47}i+3\sqrt{2}\approx 4.242640687-6.8556546i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-6\sqrt{2}.

$$x^{2}-6x\sqrt{2}+65=0$$

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.

$$x^{2}+\left(-6\sqrt{2}\right)x+65=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6\sqrt{2} mō b, me 65 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{\left(-6\sqrt{2}\right)^{2}-4\times 65}}{2}$$

Pūrua -6\sqrt{2}.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{72-4\times 65}}{2}$$

Whakareatia -4 ki te 65.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{72-260}}{2}$$

Tāpiri 72 ki te -260.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{-188}}{2}$$

Tuhia te pūtakerua o te -188.

$$x=\frac{-\left(-6\sqrt{2}\right)±2\sqrt{47}i}{2}$$

Ko te tauaro o -6\sqrt{2} ko 6\sqrt{2}.

$$x=\frac{6\sqrt{2}±2\sqrt{47}i}{2}$$

Nā, me whakaoti te whārite x=\frac{6\sqrt{2}±2\sqrt{47}i}{2} ina he tāpiri te ±. Tāpiri 6\sqrt{2} ki te 2i\sqrt{47}.

$$x=\frac{6\sqrt{2}+2\sqrt{47}i}{2}$$

Whakawehe 6\sqrt{2}+2i\sqrt{47} ki te 2.

$$x=3\sqrt{2}+\sqrt{47}i$$

Nā, me whakaoti te whārite x=\frac{6\sqrt{2}±2\sqrt{47}i}{2} ina he tango te ±. Tango 2i\sqrt{47} mai i 6\sqrt{2}.

$$x=\frac{-2\sqrt{47}i+6\sqrt{2}}{2}$$

Whakawehe 6\sqrt{2}-2i\sqrt{47} ki te 2.

$$x=-\sqrt{47}i+3\sqrt{2}$$

Kua oti te whārite te whakatau.

$$x=3\sqrt{2}+\sqrt{47}i x=-\sqrt{47}i+3\sqrt{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-6\sqrt{2}.

$$x^{2}-6x\sqrt{2}+65=0$$

Tangohia te 65 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}-6x\sqrt{2}=-65$$

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.

$$x^{2}+\left(-6\sqrt{2}\right)x=-65$$

Whakawehea te -6\sqrt{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3\sqrt{2}. Nā, tāpiria te pūrua o te -3\sqrt{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\left(-6\sqrt{2}\right)x+\left(-3\sqrt{2}\right)^{2}=-65+\left(-3\sqrt{2}\right)^{2}$$

Pūrua -3\sqrt{2}.

$$x^{2}+\left(-6\sqrt{2}\right)x+18=-65+18$$

Tāpiri -65 ki te 18.

$$x^{2}+\left(-6\sqrt{2}\right)x+18=-47$$

Tauwehea x^{2}+\left(-6\sqrt{2}\right)x+18. 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}.

$$\left(x-3\sqrt{2}\right)^{2}=-47$$

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

$$\sqrt{\left(x-3\sqrt{2}\right)^{2}}=\sqrt{-47}$$

Whakarūnātia.

$$x-3\sqrt{2}=\sqrt{47}i x-3\sqrt{2}=-\sqrt{47}i$$

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

$$x=3\sqrt{2}+\sqrt{47}i x=-\sqrt{47}i+3\sqrt{2}$$