Whakaoti i te x^2+9x+9=0

$$x ^ { 2 } + 9 x + 9 = 0$$

Whakaoti mō x

$$x=\frac{3\sqrt{5}-9}{2}\approx -1.145898034$$

$$x=\frac{-3\sqrt{5}-9}{2}\approx -7.854101966$$

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.

$$x^{2}+9x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 9 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 9}}{2}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 9}}{2}$$

Whakareatia -4 ki te 9.

$$x=\frac{-9±\sqrt{81-36}}{2}$$

Tāpiri 81 ki te -36.

$$x=\frac{-9±\sqrt{45}}{2}$$

Tuhia te pūtakerua o te 45.

$$x=\frac{-9±3\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{-9±3\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri -9 ki te 3\sqrt{5}.

$$x=\frac{3\sqrt{5}-9}{2}$$

Nā, me whakaoti te whārite x=\frac{-9±3\sqrt{5}}{2} ina he tango te ±. Tango 3\sqrt{5} mai i -9.

$$x=\frac{-3\sqrt{5}-9}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{5}-9}{2} x=\frac{-3\sqrt{5}-9}{2}$$

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.

$$x^{2}+9x+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x^{2}+9x+9-9=-9$$

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

$$x^{2}+9x=-9$$

Whakawehea te 9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{2}. Nā, tāpiria te pūrua o te \frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-9+\left(\frac{9}{2}\right)^{2}$$

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+9x+\frac{81}{4}=-9+\frac{81}{4}$$

Tāpiri -9 ki te \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{45}{4}$$

Tauwehea x^{2}+9x+\frac{81}{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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{45}{4}$$

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

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Whakarūnātia.

$$x+\frac{9}{2}=\frac{3\sqrt{5}}{2} x+\frac{9}{2}=-\frac{3\sqrt{5}}{2}$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{5}-9}{2} x=\frac{-3\sqrt{5}-9}{2}$$