Whakaoti i te x^2-15x+100=0

$$x ^ { 2 } - 15 x + 100 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{15+5\sqrt{7}i}{2}\approx 7.5+6.614378278i$$

$$x=\frac{-5\sqrt{7}i+15}{2}\approx 7.5-6.614378278i$$

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.

$$x^{2}-15x+100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -15 mō b, me 100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 100}}{2}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 100}}{2}$$

Whakareatia -4 ki te 100.

$$x=\frac{-\left(-15\right)±\sqrt{225-400}}{2}$$

Tāpiri 225 ki te -400.

$$x=\frac{-\left(-15\right)±\sqrt{-175}}{2}$$

Tuhia te pūtakerua o te -175.

$$x=\frac{-\left(-15\right)±5\sqrt{7}i}{2}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±5\sqrt{7}i}{2}$$

Nā, me whakaoti te whārite x=\frac{15±5\sqrt{7}i}{2} ina he tāpiri te ±. Tāpiri 15 ki te 5i\sqrt{7}.

$$x=\frac{15+5\sqrt{7}i}{2}$$

Nā, me whakaoti te whārite x=\frac{15±5\sqrt{7}i}{2} ina he tango te ±. Tango 5i\sqrt{7} mai i 15.

$$x=\frac{-5\sqrt{7}i+15}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{15+5\sqrt{7}i}{2} x=\frac{-5\sqrt{7}i+15}{2}$$

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.

$$x^{2}-15x+100=0$$

Me tango 100 mai i ngā taha e rua o te whārite.

$$x^{2}-15x+100-100=-100$$

Mā te tango i te 100 i a ia ake anō ka toe ko te 0.

$$x^{2}-15x=-100$$

Whakawehea te -15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{2}. Nā, tāpiria te pūrua o te -\frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-100+\left(-\frac{15}{2}\right)^{2}$$

Pūruatia -\frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-15x+\frac{225}{4}=-100+\frac{225}{4}$$

Tāpiri -100 ki te \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=-\frac{175}{4}$$

Tauwehea x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=-\frac{175}{4}$$

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

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{-\frac{175}{4}}$$

Whakarūnātia.

$$x-\frac{15}{2}=\frac{5\sqrt{7}i}{2} x-\frac{15}{2}=-\frac{5\sqrt{7}i}{2}$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$x=\frac{15+5\sqrt{7}i}{2} x=\frac{-5\sqrt{7}i+15}{2}$$