Whakaoti i te 7k^2+18k-27=0

$$7 k ^ { 2 } + 18 k - 27 = 0$$

Whakaoti mō k

$$k = \frac{3 \sqrt{30} - 9}{7} \approx 1.061668104$$

$$k=\frac{-3\sqrt{30}-9}{7}\approx -3.633096675$$

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.

$$7k^{2}+18k-27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 18 mō b, me -27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-18±\sqrt{18^{2}-4\times 7\left(-27\right)}}{2\times 7}$$

Pūrua 18.

$$k=\frac{-18±\sqrt{324-4\times 7\left(-27\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$k=\frac{-18±\sqrt{324-28\left(-27\right)}}{2\times 7}$$

Whakareatia -28 ki te -27.

$$k=\frac{-18±\sqrt{324+756}}{2\times 7}$$

Tāpiri 324 ki te 756.

$$k=\frac{-18±\sqrt{1080}}{2\times 7}$$

Tuhia te pūtakerua o te 1080.

$$k=\frac{-18±6\sqrt{30}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$k=\frac{-18±6\sqrt{30}}{14}$$

Nā, me whakaoti te whārite k=\frac{-18±6\sqrt{30}}{14} ina he tāpiri te ±. Tāpiri -18 ki te 6\sqrt{30}.

$$k=\frac{6\sqrt{30}-18}{14}$$

Whakawehe -18+6\sqrt{30} ki te 14.

$$k=\frac{3\sqrt{30}-9}{7}$$

Nā, me whakaoti te whārite k=\frac{-18±6\sqrt{30}}{14} ina he tango te ±. Tango 6\sqrt{30} mai i -18.

$$k=\frac{-6\sqrt{30}-18}{14}$$

Whakawehe -18-6\sqrt{30} ki te 14.

$$k=\frac{-3\sqrt{30}-9}{7}$$

Kua oti te whārite te whakatau.

$$k=\frac{3\sqrt{30}-9}{7} k=\frac{-3\sqrt{30}-9}{7}$$

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.

$$7k^{2}+18k-27=0$$

Me tāpiri 27 ki ngā taha e rua o te whārite.

$$7k^{2}+18k-27-\left(-27\right)=-\left(-27\right)$$

Mā te tango i te -27 i a ia ake anō ka toe ko te 0.

$$7k^{2}+18k=-\left(-27\right)$$

Tango -27 mai i 0.

$$7k^{2}+18k=27$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7k^{2}+18k}{7}=\frac{27}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$k^{2}+\frac{18}{7}k=\frac{27}{7}$$

Whakawehea te \frac{18}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{7}. Nā, tāpiria te pūrua o te \frac{9}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}+\frac{18}{7}k+\left(\frac{9}{7}\right)^{2}=\frac{27}{7}+\left(\frac{9}{7}\right)^{2}$$

Pūruatia \frac{9}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$k^{2}+\frac{18}{7}k+\frac{81}{49}=\frac{27}{7}+\frac{81}{49}$$

Tāpiri \frac{27}{7} ki te \frac{81}{49} 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.

$$k^{2}+\frac{18}{7}k+\frac{81}{49}=\frac{270}{49}$$

Tauwehea k^{2}+\frac{18}{7}k+\frac{81}{49}. 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}.

$$\left(k+\frac{9}{7}\right)^{2}=\frac{270}{49}$$

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

$$\sqrt{\left(k+\frac{9}{7}\right)^{2}}=\sqrt{\frac{270}{49}}$$

Whakarūnātia.

$$k+\frac{9}{7}=\frac{3\sqrt{30}}{7} k+\frac{9}{7}=-\frac{3\sqrt{30}}{7}$$

Me tango \frac{9}{7} mai i ngā taha e rua o te whārite.

$$k=\frac{3\sqrt{30}-9}{7} k=\frac{-3\sqrt{30}-9}{7}$$