Whakaoti i te 2x^2+9x-1=0

$$2 x ^ { 2 } + 9 x - 1 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{89}-9}{4}\approx 0.108495283$$

$$x=\frac{-\sqrt{89}-9}{4}\approx -4.608495283$$

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.

$$2x^{2}+9x-1=0$$

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

$$x=\frac{-9±\sqrt{9^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 2\left(-1\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-9±\sqrt{81-8\left(-1\right)}}{2\times 2}$$

Whakareatia -8 ki te -1.

$$x=\frac{-9±\sqrt{81+8}}{2\times 2}$$

Tāpiri 81 ki te 8.

$$x=\frac{-9±\sqrt{89}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-9±\sqrt{89}}{4}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{89}}{4} ina he tāpiri te ±. Tāpiri -9 ki te \sqrt{89}.

$$x=\frac{\sqrt{89}-9}{4}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{89}}{4} ina he tango te ±. Tango \sqrt{89} mai i -9.

$$x=\frac{-\sqrt{89}-9}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{89}-9}{4} x=\frac{-\sqrt{89}-9}{4}$$

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.

$$2x^{2}+9x-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$2x^{2}+9x-1-\left(-1\right)=-\left(-1\right)$$

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

$$2x^{2}+9x=-\left(-1\right)$$

Tango -1 mai i 0.

$$2x^{2}+9x=1$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+9x}{2}=\frac{1}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{9}{2}x=\frac{1}{2}$$

Whakawehea te \frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{4}. Nā, tāpiria te pūrua o te \frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{9}{2}x+\left(\frac{9}{4}\right)^{2}=\frac{1}{2}+\left(\frac{9}{4}\right)^{2}$$

Pūruatia \frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{1}{2}+\frac{81}{16}$$

Tāpiri \frac{1}{2} ki te \frac{81}{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.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{89}{16}$$

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

$$\left(x+\frac{9}{4}\right)^{2}=\frac{89}{16}$$

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

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{89}{16}}$$

Whakarūnātia.

$$x+\frac{9}{4}=\frac{\sqrt{89}}{4} x+\frac{9}{4}=-\frac{\sqrt{89}}{4}$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{89}-9}{4} x=\frac{-\sqrt{89}-9}{4}$$