Whakaoti i te x-x^2=5/18

$$x - x ^ { 2 } = \frac { 5 } { 18 }$$

Whakaoti mō x (complex solution)

$$x=\frac{1}{2}+\frac{1}{6}i\approx 0.5+0.166666667i$$

$$x=\frac{1}{2}-\frac{1}{6}i\approx 0.5-0.166666667i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$-x^{2}+x=\frac{5}{18}$$

Me tango \frac{5}{18} mai i ngā taha e rua o te whārite.

$$-x^{2}+x-\frac{5}{18}=\frac{5}{18}-\frac{5}{18}$$

Mā te tango i te \frac{5}{18} i a ia ake anō ka toe ko te 0.

$$-x^{2}+x-\frac{5}{18}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 1 mō b, me -\frac{5}{18} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\left(-\frac{5}{18}\right)}}{2\left(-1\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\left(-\frac{5}{18}\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-1±\sqrt{1+4\left(-\frac{5}{18}\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -\frac{5}{18}.

$$x=\frac{-1±\sqrt{1-\frac{10}{9}}}{2\left(-1\right)}$$

Tāpiri 1 ki te -\frac{10}{9}.

$$x=\frac{-1±\sqrt{-\frac{1}{9}}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -\frac{1}{9}.

$$x=\frac{-1±\frac{1}{3}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-1±\frac{1}{3}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{1}{3}i}{-2} ina he tāpiri te ±. Tāpiri -1 ki te \frac{1}{3}i.

$$x=\frac{-1+\frac{1}{3}i}{-2}$$

Whakawehe -1+\frac{1}{3}i ki te -2.

$$x=\frac{1}{2}-\frac{1}{6}i$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{1}{3}i}{-2} ina he tango te ±. Tango \frac{1}{3}i mai i -1.

$$x=\frac{-1-\frac{1}{3}i}{-2}$$

Whakawehe -1-\frac{1}{3}i ki te -2.

$$x=\frac{1}{2}+\frac{1}{6}i$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2}-\frac{1}{6}i x=\frac{1}{2}+\frac{1}{6}i$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$-x^{2}+x=\frac{5}{18}$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+x}{-1}=\frac{\frac{5}{18}}{-1}$$

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

$$x^{2}+\frac{1}{-1}x=\frac{\frac{5}{18}}{-1}$$

Whakawehe 1 ki te -1.

$$x^{2}-x=\frac{\frac{5}{18}}{-1}$$

Whakawehe \frac{5}{18} ki te -1.

$$x^{2}-x=-\frac{5}{18}$$

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{5}{18}+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=-\frac{5}{18}+\frac{1}{4}$$

Tāpiri -\frac{5}{18} ki te \frac{1}{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.

$$x^{2}-x+\frac{1}{4}=-\frac{1}{36}$$

Tauwehea x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{1}{36}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{1}{36}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{1}{6}i x-\frac{1}{2}=-\frac{1}{6}i$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=\frac{1}{2}+\frac{1}{6}i x=\frac{1}{2}-\frac{1}{6}i$$