Whakaoti i te 3x^2+2x-3=0

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

Whakaoti mō x

$$x=\frac{\sqrt{10}-1}{3}\approx 0.72075922$$

$$x=\frac{-\sqrt{10}-1}{3}\approx -1.387425887$$

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.

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

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

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

Pūrua 2.

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

Whakareatia -4 ki te 3.

$$x=\frac{-2±\sqrt{4-12\left(-3\right)}}{2\times 3}$$

Whakareatia -12 ki te -3.

$$x=\frac{-2±\sqrt{4+36}}{2\times 3}$$

Tāpiri 4 ki te 36.

$$x=\frac{-2±\sqrt{40}}{2\times 3}$$

Tuhia te pūtakerua o te 40.

$$x=\frac{-2±2\sqrt{10}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-2±2\sqrt{10}}{6}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{10}}{6} ina he tāpiri te ±. Tāpiri -2 ki te 2\sqrt{10}.

$$x=\frac{2\sqrt{10}-2}{6}$$

Whakawehe -2+2\sqrt{10} ki te 6.

$$x=\frac{\sqrt{10}-1}{3}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{10}}{6} ina he tango te ±. Tango 2\sqrt{10} mai i -2.

$$x=\frac{-2\sqrt{10}-2}{6}$$

Whakawehe -2-2\sqrt{10} ki te 6.

$$x=\frac{-\sqrt{10}-1}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{10}-1}{3} x=\frac{-\sqrt{10}-1}{3}$$

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.

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

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

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

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

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

Tango -3 mai i 0.

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

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakawehe 3 ki te 3.

$$x^{2}+\frac{2}{3}x=1$$

Whakawehea te \frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{3}. Nā, tāpiria te pūrua o te \frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=1+\left(\frac{1}{3}\right)^{2}$$

Pūruatia \frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=1+\frac{1}{9}$$

Tāpiri 1 ki te \frac{1}{9}.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{10}{9}$$

Tauwehea te x^{2}+\frac{2}{3}x+\frac{1}{9}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{10}{9}$$

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

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{10}{9}}$$

Whakarūnātia.

$$x+\frac{1}{3}=\frac{\sqrt{10}}{3} x+\frac{1}{3}=-\frac{\sqrt{10}}{3}$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{10}-1}{3} x=\frac{-\sqrt{10}-1}{3}$$