Whakaoti i te 7f^2+7f-9=0

$$7 f ^ { 2 } + 7 f - 9 = 0$$

Whakaoti mō f

$$f=\frac{\sqrt{301}}{14}-\frac{1}{2}\approx 0.739239398$$

$$f=-\frac{\sqrt{301}}{14}-\frac{1}{2}\approx -1.739239398$$

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.

$$7f^{2}+7f-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 7 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$f=\frac{-7±\sqrt{7^{2}-4\times 7\left(-9\right)}}{2\times 7}$$

Pūrua 7.

$$f=\frac{-7±\sqrt{49-4\times 7\left(-9\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$f=\frac{-7±\sqrt{49-28\left(-9\right)}}{2\times 7}$$

Whakareatia -28 ki te -9.

$$f=\frac{-7±\sqrt{49+252}}{2\times 7}$$

Tāpiri 49 ki te 252.

$$f=\frac{-7±\sqrt{301}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$f=\frac{-7±\sqrt{301}}{14}$$

Nā, me whakaoti te whārite f=\frac{-7±\sqrt{301}}{14} ina he tāpiri te ±. Tāpiri -7 ki te \sqrt{301}.

$$f=\frac{\sqrt{301}-7}{14}$$

Whakawehe -7+\sqrt{301} ki te 14.

$$f=\frac{\sqrt{301}}{14}-\frac{1}{2}$$

Nā, me whakaoti te whārite f=\frac{-7±\sqrt{301}}{14} ina he tango te ±. Tango \sqrt{301} mai i -7.

$$f=\frac{-\sqrt{301}-7}{14}$$

Whakawehe -7-\sqrt{301} ki te 14.

$$f=-\frac{\sqrt{301}}{14}-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$f=\frac{\sqrt{301}}{14}-\frac{1}{2} f=-\frac{\sqrt{301}}{14}-\frac{1}{2}$$

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.

$$7f^{2}+7f-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$7f^{2}+7f-9-\left(-9\right)=-\left(-9\right)$$

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

$$7f^{2}+7f=-\left(-9\right)$$

Tango -9 mai i 0.

$$7f^{2}+7f=9$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7f^{2}+7f}{7}=\frac{9}{7}$$

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

$$f^{2}+\frac{7}{7}f=\frac{9}{7}$$

Whakawehe 7 ki te 7.

$$f^{2}+f=\frac{9}{7}$$

Whakawehea te 1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{2}. Nā, tāpiria te pūrua o te \frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$f^{2}+f+\left(\frac{1}{2}\right)^{2}=\frac{9}{7}+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$f^{2}+f+\frac{1}{4}=\frac{9}{7}+\frac{1}{4}$$

Tāpiri \frac{9}{7} ki te \frac{1}{4} 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.

$$f^{2}+f+\frac{1}{4}=\frac{43}{28}$$

Tauwehea f^{2}+f+\frac{1}{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}.

$$\left(f+\frac{1}{2}\right)^{2}=\frac{43}{28}$$

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

$$\sqrt{\left(f+\frac{1}{2}\right)^{2}}=\sqrt{\frac{43}{28}}$$

Whakarūnātia.

$$f+\frac{1}{2}=\frac{\sqrt{301}}{14} f+\frac{1}{2}=-\frac{\sqrt{301}}{14}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$f=\frac{\sqrt{301}}{14}-\frac{1}{2} f=-\frac{\sqrt{301}}{14}-\frac{1}{2}$$