Whakaoti i te 3x^2-20x+1=0

$$3 x ^ { 2 } - 20 x + 1 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{97} + 10}{3} \approx 6.616285934$$

$$x=\frac{10-\sqrt{97}}{3}\approx 0.050380733$$

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.

$$3x^{2}-20x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -20 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 3}}{2\times 3}$$

Pūrua -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 3}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-20\right)±\sqrt{400-12}}{2\times 3}$$

Tāpiri 400 ki te -12.

$$x=\frac{-\left(-20\right)±\sqrt{388}}{2\times 3}$$

Tuhia te pūtakerua o te 388.

$$x=\frac{-\left(-20\right)±2\sqrt{97}}{2\times 3}$$

Ko te tauaro o -20 ko 20.

$$x=\frac{20±2\sqrt{97}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{20±2\sqrt{97}}{6}$$

Nā, me whakaoti te whārite x=\frac{20±2\sqrt{97}}{6} ina he tāpiri te ±. Tāpiri 20 ki te 2\sqrt{97}.

$$x=\frac{2\sqrt{97}+20}{6}$$

Whakawehe 20+2\sqrt{97} ki te 6.

$$x=\frac{\sqrt{97}+10}{3}$$

Nā, me whakaoti te whārite x=\frac{20±2\sqrt{97}}{6} ina he tango te ±. Tango 2\sqrt{97} mai i 20.

$$x=\frac{20-2\sqrt{97}}{6}$$

Whakawehe 20-2\sqrt{97} ki te 6.

$$x=\frac{10-\sqrt{97}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{97}+10}{3} x=\frac{10-\sqrt{97}}{3}$$

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.

$$3x^{2}-20x+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$3x^{2}-20x+1-1=-1$$

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

$$3x^{2}-20x=-1$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-20x}{3}=-\frac{1}{3}$$

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

$$x^{2}-\frac{20}{3}x=-\frac{1}{3}$$

Whakawehea te -\frac{20}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{10}{3}. Nā, tāpiria te pūrua o te -\frac{10}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{20}{3}x+\left(-\frac{10}{3}\right)^{2}=-\frac{1}{3}+\left(-\frac{10}{3}\right)^{2}$$

Pūruatia -\frac{10}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=-\frac{1}{3}+\frac{100}{9}$$

Tāpiri -\frac{1}{3} ki te \frac{100}{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.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=\frac{97}{9}$$

Tauwehea x^{2}-\frac{20}{3}x+\frac{100}{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}.

$$\left(x-\frac{10}{3}\right)^{2}=\frac{97}{9}$$

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

$$\sqrt{\left(x-\frac{10}{3}\right)^{2}}=\sqrt{\frac{97}{9}}$$

Whakarūnātia.

$$x-\frac{10}{3}=\frac{\sqrt{97}}{3} x-\frac{10}{3}=-\frac{\sqrt{97}}{3}$$

Me tāpiri \frac{10}{3} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{97}+10}{3} x=\frac{10-\sqrt{97}}{3}$$