Whakaoti i te 6p^2+9p-37=0

$$6 p ^ { 2 } + 9 p - 37 = 0$$

Whakaoti mō p

$$p=\frac{\sqrt{969}}{12}-\frac{3}{4}\approx 1.844063736$$

$$p=-\frac{\sqrt{969}}{12}-\frac{3}{4}\approx -3.344063736$$

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.

$$6p^{2}+9p-37=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 9 mō b, me -37 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-9±\sqrt{9^{2}-4\times 6\left(-37\right)}}{2\times 6}$$

Pūrua 9.

$$p=\frac{-9±\sqrt{81-4\times 6\left(-37\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$p=\frac{-9±\sqrt{81-24\left(-37\right)}}{2\times 6}$$

Whakareatia -24 ki te -37.

$$p=\frac{-9±\sqrt{81+888}}{2\times 6}$$

Tāpiri 81 ki te 888.

$$p=\frac{-9±\sqrt{969}}{2\times 6}$$

Whakareatia 2 ki te 6.

$$p=\frac{-9±\sqrt{969}}{12}$$

Nā, me whakaoti te whārite p=\frac{-9±\sqrt{969}}{12} ina he tāpiri te ±. Tāpiri -9 ki te \sqrt{969}.

$$p=\frac{\sqrt{969}-9}{12}$$

Whakawehe -9+\sqrt{969} ki te 12.

$$p=\frac{\sqrt{969}}{12}-\frac{3}{4}$$

Nā, me whakaoti te whārite p=\frac{-9±\sqrt{969}}{12} ina he tango te ±. Tango \sqrt{969} mai i -9.

$$p=\frac{-\sqrt{969}-9}{12}$$

Whakawehe -9-\sqrt{969} ki te 12.

$$p=-\frac{\sqrt{969}}{12}-\frac{3}{4}$$

Kua oti te whārite te whakatau.

$$p=\frac{\sqrt{969}}{12}-\frac{3}{4} p=-\frac{\sqrt{969}}{12}-\frac{3}{4}$$

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.

$$6p^{2}+9p-37=0$$

Me tāpiri 37 ki ngā taha e rua o te whārite.

$$6p^{2}+9p-37-\left(-37\right)=-\left(-37\right)$$

Mā te tango i te -37 i a ia ake anō ka toe ko te 0.

$$6p^{2}+9p=-\left(-37\right)$$

Tango -37 mai i 0.

$$6p^{2}+9p=37$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6p^{2}+9p}{6}=\frac{37}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$p^{2}+\frac{9}{6}p=\frac{37}{6}$$

Whakahekea te hautanga \frac{9}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$p^{2}+\frac{3}{2}p=\frac{37}{6}$$

Whakawehea te \frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{4}. Nā, tāpiria te pūrua o te \frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$p^{2}+\frac{3}{2}p+\left(\frac{3}{4}\right)^{2}=\frac{37}{6}+\left(\frac{3}{4}\right)^{2}$$

Pūruatia \frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$p^{2}+\frac{3}{2}p+\frac{9}{16}=\frac{37}{6}+\frac{9}{16}$$

Tāpiri \frac{37}{6} ki te \frac{9}{16} 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.

$$p^{2}+\frac{3}{2}p+\frac{9}{16}=\frac{323}{48}$$

Tauwehea p^{2}+\frac{3}{2}p+\frac{9}{16}. 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}.

$$\left(p+\frac{3}{4}\right)^{2}=\frac{323}{48}$$

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

$$\sqrt{\left(p+\frac{3}{4}\right)^{2}}=\sqrt{\frac{323}{48}}$$

Whakarūnātia.

$$p+\frac{3}{4}=\frac{\sqrt{969}}{12} p+\frac{3}{4}=-\frac{\sqrt{969}}{12}$$

Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.

$$p=\frac{\sqrt{969}}{12}-\frac{3}{4} p=-\frac{\sqrt{969}}{12}-\frac{3}{4}$$