Whakaoti i te x^2+7x+47=0

$$x ^ { 2 } + 7 x + 47 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-7+\sqrt{139}i}{2}\approx -3.5+5.894913061i$$

$$x=\frac{-\sqrt{139}i-7}{2}\approx -3.5-5.894913061i$$

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.

$$x^{2}+7x+47=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 7 mō b, me 47 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 47}}{2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 47}}{2}$$

Whakareatia -4 ki te 47.

$$x=\frac{-7±\sqrt{49-188}}{2}$$

Tāpiri 49 ki te -188.

$$x=\frac{-7±\sqrt{-139}}{2}$$

Tuhia te pūtakerua o te -139.

$$x=\frac{-7±\sqrt{139}i}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{139}i}{2} ina he tāpiri te ±. Tāpiri -7 ki te i\sqrt{139}.

$$x=\frac{-7+\sqrt{139}i}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{139}i}{2} ina he tango te ±. Tango i\sqrt{139} mai i -7.

$$x=\frac{-\sqrt{139}i-7}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-7+\sqrt{139}i}{2} x=\frac{-\sqrt{139}i-7}{2}$$

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.

$$x^{2}+7x+47=0$$

Me tango 47 mai i ngā taha e rua o te whārite.

$$x^{2}+7x+47-47=-47$$

Mā te tango i te 47 i a ia ake anō ka toe ko te 0.

$$x^{2}+7x=-47$$

Whakawehea te 7, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{2}. Nā, tāpiria te pūrua o te \frac{7}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-47+\left(\frac{7}{2}\right)^{2}$$

Pūruatia \frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+7x+\frac{49}{4}=-47+\frac{49}{4}$$

Tāpiri -47 ki te \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=-\frac{139}{4}$$

Tauwehea x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=-\frac{139}{4}$$

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

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{-\frac{139}{4}}$$

Whakarūnātia.

$$x+\frac{7}{2}=\frac{\sqrt{139}i}{2} x+\frac{7}{2}=-\frac{\sqrt{139}i}{2}$$

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{-7+\sqrt{139}i}{2} x=\frac{-\sqrt{139}i-7}{2}$$