Whakaoti i te -3k^2-18k+57=0

$$- 3 k ^ { 2 } - 18 k + 57 = 0$$

Whakaoti mō k

$$k=2\sqrt{7}-3\approx 2.291502622$$

$$k=-2\sqrt{7}-3\approx -8.291502622$$

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.

$$-3k^{2}-18k+57=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, -18 mō b, me 57 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-3\right)\times 57}}{2\left(-3\right)}$$

Pūrua -18.

$$k=\frac{-\left(-18\right)±\sqrt{324-4\left(-3\right)\times 57}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$k=\frac{-\left(-18\right)±\sqrt{324+12\times 57}}{2\left(-3\right)}$$

Whakareatia 12 ki te 57.

$$k=\frac{-\left(-18\right)±\sqrt{324+684}}{2\left(-3\right)}$$

Tāpiri 324 ki te 684.

$$k=\frac{-\left(-18\right)±\sqrt{1008}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 1008.

$$k=\frac{-\left(-18\right)±12\sqrt{7}}{2\left(-3\right)}$$

Ko te tauaro o -18 ko 18.

$$k=\frac{18±12\sqrt{7}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$k=\frac{18±12\sqrt{7}}{-6}$$

Nā, me whakaoti te whārite k=\frac{18±12\sqrt{7}}{-6} ina he tāpiri te ±. Tāpiri 18 ki te 12\sqrt{7}.

$$k=\frac{12\sqrt{7}+18}{-6}$$

Whakawehe 18+12\sqrt{7} ki te -6.

$$k=-2\sqrt{7}-3$$

Nā, me whakaoti te whārite k=\frac{18±12\sqrt{7}}{-6} ina he tango te ±. Tango 12\sqrt{7} mai i 18.

$$k=\frac{18-12\sqrt{7}}{-6}$$

Whakawehe 18-12\sqrt{7} ki te -6.

$$k=2\sqrt{7}-3$$

Kua oti te whārite te whakatau.

$$k=-2\sqrt{7}-3 k=2\sqrt{7}-3$$

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.

$$-3k^{2}-18k+57=0$$

Me tango 57 mai i ngā taha e rua o te whārite.

$$-3k^{2}-18k+57-57=-57$$

Mā te tango i te 57 i a ia ake anō ka toe ko te 0.

$$-3k^{2}-18k=-57$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3k^{2}-18k}{-3}=-\frac{57}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$k^{2}+\left(-\frac{18}{-3}\right)k=-\frac{57}{-3}$$

Whakawehe -18 ki te -3.

$$k^{2}+6k=-\frac{57}{-3}$$

Whakawehe -57 ki te -3.

$$k^{2}+6k=19$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}+6k+3^{2}=19+3^{2}$$

Pūrua 3.

$$k^{2}+6k+9=19+9$$

Tāpiri 19 ki te 9.

$$k^{2}+6k+9=28$$

Tauwehea k^{2}+6k+9. 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}.

$$\left(k+3\right)^{2}=28$$

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

$$\sqrt{\left(k+3\right)^{2}}=\sqrt{28}$$

Whakarūnātia.

$$k+3=2\sqrt{7} k+3=-2\sqrt{7}$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$k=2\sqrt{7}-3 k=-2\sqrt{7}-3$$