Whakaoti i te 3x^2+8x-3=65

$$3 { x }^{ 2 } +8x-3=65$$

Whakaoti mō x

$$x = \frac{2 \sqrt{55} - 4}{3} \approx 3.610798991$$

$$x=\frac{-2\sqrt{55}-4}{3}\approx -6.277465658$$

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.

$$3x^{2}+8x-3=65$$

Me tango 65 mai i ngā taha e rua o te whārite.

$$3x^{2}+8x-3-65=65-65$$

Mā te tango i te 65 i a ia ake anō ka toe ko te 0.

$$3x^{2}+8x-3-65=0$$

Tango 65 mai i -3.

$$3x^{2}+8x-68=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 8 mō b, me -68 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 3\left(-68\right)}}{2\times 3}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 3\left(-68\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-8±\sqrt{64-12\left(-68\right)}}{2\times 3}$$

Whakareatia -12 ki te -68.

$$x=\frac{-8±\sqrt{64+816}}{2\times 3}$$

Tāpiri 64 ki te 816.

$$x=\frac{-8±\sqrt{880}}{2\times 3}$$

Tuhia te pūtakerua o te 880.

$$x=\frac{-8±4\sqrt{55}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-8±4\sqrt{55}}{6}$$

Nā, me whakaoti te whārite x=\frac{-8±4\sqrt{55}}{6} ina he tāpiri te ±. Tāpiri -8 ki te 4\sqrt{55}.

$$x=\frac{4\sqrt{55}-8}{6}$$

Whakawehe -8+4\sqrt{55} ki te 6.

$$x=\frac{2\sqrt{55}-4}{3}$$

Nā, me whakaoti te whārite x=\frac{-8±4\sqrt{55}}{6} ina he tango te ±. Tango 4\sqrt{55} mai i -8.

$$x=\frac{-4\sqrt{55}-8}{6}$$

Whakawehe -8-4\sqrt{55} ki te 6.

$$x=\frac{-2\sqrt{55}-4}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{55}-4}{3} x=\frac{-2\sqrt{55}-4}{3}$$

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.

$$3x^{2}+8x-3=65$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$3x^{2}+8x-3-\left(-3\right)=65-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$3x^{2}+8x=65-\left(-3\right)$$

Tango -3 mai i 65.

$$3x^{2}+8x=68$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+8x}{3}=\frac{68}{3}$$

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

$$x^{2}+\frac{8}{3}x=\frac{68}{3}$$

Whakawehea te \frac{8}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{4}{3}. Nā, tāpiria te pūrua o te \frac{4}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=\frac{68}{3}+\left(\frac{4}{3}\right)^{2}$$

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

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{68}{3}+\frac{16}{9}$$

Tāpiri \frac{68}{3} ki te \frac{16}{9} 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.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{220}{9}$$

Tauwehea x^{2}+\frac{8}{3}x+\frac{16}{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}.

$$\left(x+\frac{4}{3}\right)^{2}=\frac{220}{9}$$

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

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{\frac{220}{9}}$$

Whakarūnātia.

$$x+\frac{4}{3}=\frac{2\sqrt{55}}{3} x+\frac{4}{3}=-\frac{2\sqrt{55}}{3}$$

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

$$x=\frac{2\sqrt{55}-4}{3} x=\frac{-2\sqrt{55}-4}{3}$$