Whakaoti i te 25x^2-90x+77=0

$$25 { x }^{ 2 } -90x+77=0$$

Whakaoti mō x

$$x = \frac{7}{5} = 1\frac{2}{5} = 1.4$$

$$x = \frac{11}{5} = 2\frac{1}{5} = 2.2$$

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.

$$25x^{2}-90x+77=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, -90 mō b, me 77 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 25\times 77}}{2\times 25}$$

Pūrua -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 25\times 77}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-\left(-90\right)±\sqrt{8100-100\times 77}}{2\times 25}$$

Whakareatia -100 ki te 77.

$$x=\frac{-\left(-90\right)±\sqrt{8100-7700}}{2\times 25}$$

Tāpiri 8100 ki te -7700.

$$x=\frac{-\left(-90\right)±\sqrt{400}}{2\times 25}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-90\right)±20}{2\times 25}$$

Ko te tauaro o -90 ko 90.

$$x=\frac{90±20}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{90±20}{50}$$

Nā, me whakaoti te whārite x=\frac{90±20}{50} ina he tāpiri te ±. Tāpiri 90 ki te 20.

$$x=\frac{110}{50}$$

Whakahekea te hautanga \frac{110}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{11}{5}$$

Nā, me whakaoti te whārite x=\frac{90±20}{50} ina he tango te ±. Tango 20 mai i 90.

$$x=\frac{70}{50}$$

Whakahekea te hautanga \frac{70}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{7}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{11}{5} x=\frac{7}{5}$$

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.

$$25x^{2}-90x+77=0$$

Me tango 77 mai i ngā taha e rua o te whārite.

$$25x^{2}-90x+77-77=-77$$

Mā te tango i te 77 i a ia ake anō ka toe ko te 0.

$$25x^{2}-90x=-77$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}-90x}{25}=-\frac{77}{25}$$

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

$$x^{2}+\left(-\frac{90}{25}\right)x=-\frac{77}{25}$$

Whakahekea te hautanga \frac{-90}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{18}{5}x=-\frac{77}{25}$$

Whakawehea te -\frac{18}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{5}. Nā, tāpiria te pūrua o te -\frac{9}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{18}{5}x+\left(-\frac{9}{5}\right)^{2}=-\frac{77}{25}+\left(-\frac{9}{5}\right)^{2}$$

Pūruatia -\frac{9}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=\frac{-77+81}{25}$$

Tāpiri -\frac{77}{25} ki te \frac{81}{25} 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.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=\frac{4}{25}$$

Tauwehea x^{2}-\frac{18}{5}x+\frac{81}{25}. 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}.

$$\left(x-\frac{9}{5}\right)^{2}=\frac{4}{25}$$

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

$$\sqrt{\left(x-\frac{9}{5}\right)^{2}}=\sqrt{\frac{4}{25}}$$

Whakarūnātia.

$$x-\frac{9}{5}=\frac{2}{5} x-\frac{9}{5}=-\frac{2}{5}$$

Me tāpiri \frac{9}{5} ki ngā taha e rua o te whārite.

$$x=\frac{11}{5} x=\frac{7}{5}$$