Whakaoti i te 25x^2-19x-3=0

$$25 x ^ { 2 } - 19 x - 3 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{661}+19}{50}\approx 0.894198405$$

$$x=\frac{19-\sqrt{661}}{50}\approx -0.134198405$$

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.

$$25x^{2}-19x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, -19 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 25\left(-3\right)}}{2\times 25}$$

Pūrua -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 25\left(-3\right)}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-\left(-19\right)±\sqrt{361-100\left(-3\right)}}{2\times 25}$$

Whakareatia -100 ki te -3.

$$x=\frac{-\left(-19\right)±\sqrt{361+300}}{2\times 25}$$

Tāpiri 361 ki te 300.

$$x=\frac{-\left(-19\right)±\sqrt{661}}{2\times 25}$$

Ko te tauaro o -19 ko 19.

$$x=\frac{19±\sqrt{661}}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{19±\sqrt{661}}{50}$$

Nā, me whakaoti te whārite x=\frac{19±\sqrt{661}}{50} ina he tāpiri te ±. Tāpiri 19 ki te \sqrt{661}.

$$x=\frac{\sqrt{661}+19}{50}$$

Nā, me whakaoti te whārite x=\frac{19±\sqrt{661}}{50} ina he tango te ±. Tango \sqrt{661} mai i 19.

$$x=\frac{19-\sqrt{661}}{50}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{661}+19}{50} x=\frac{19-\sqrt{661}}{50}$$

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.

$$25x^{2}-19x-3=0$$

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

$$25x^{2}-19x-3-\left(-3\right)=-\left(-3\right)$$

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

$$25x^{2}-19x=-\left(-3\right)$$

Tango -3 mai i 0.

$$25x^{2}-19x=3$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}-19x}{25}=\frac{3}{25}$$

Mā te whakawehe ki te 25 ka wetekia te whakareanga ki te 25.

$$x^{2}-\frac{19}{25}x=\frac{3}{25}$$

Whakawehea te -\frac{19}{25}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{50}. Nā, tāpiria te pūrua o te -\frac{19}{50} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{19}{25}x+\left(-\frac{19}{50}\right)^{2}=\frac{3}{25}+\left(-\frac{19}{50}\right)^{2}$$

Pūruatia -\frac{19}{50} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{19}{25}x+\frac{361}{2500}=\frac{3}{25}+\frac{361}{2500}$$

Tāpiri \frac{3}{25} ki te \frac{361}{2500} 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.

$$x^{2}-\frac{19}{25}x+\frac{361}{2500}=\frac{661}{2500}$$

Tauwehea te x^{2}-\frac{19}{25}x+\frac{361}{2500}. 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}.

$$\left(x-\frac{19}{50}\right)^{2}=\frac{661}{2500}$$

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

$$\sqrt{\left(x-\frac{19}{50}\right)^{2}}=\sqrt{\frac{661}{2500}}$$

Whakarūnātia.

$$x-\frac{19}{50}=\frac{\sqrt{661}}{50} x-\frac{19}{50}=-\frac{\sqrt{661}}{50}$$

Me tāpiri \frac{19}{50} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{661}+19}{50} x=\frac{19-\sqrt{661}}{50}$$