Whakaoti i te 5x^2-70x+245=0

$$5 x ^ { 2 } - 70 x + 245 = 0$$

Whakaoti mō x

$$x=7$$

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.

$$5x^{2}-70x+245=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -70 mō b, me 245 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 5\times 245}}{2\times 5}$$

Pūrua -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 5\times 245}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-70\right)±\sqrt{4900-20\times 245}}{2\times 5}$$

Whakareatia -20 ki te 245.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4900}}{2\times 5}$$

Tāpiri 4900 ki te -4900.

$$x=\frac{-\left(-70\right)±\sqrt{0}}{2\times 5}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-70}{2\times 5}$$

Ko te tauaro o -70 ko 70.

$$x=\frac{70}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{70}{10}$$

Whakawehe 70 ki te 10.

$$x=7$$

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.

$$5x^{2}-70x+245=0$$

Me tango 245 mai i ngā taha e rua o te whārite.

$$5x^{2}-70x+245-245=-245$$

Mā te tango i te 245 i a ia ake anō ka toe ko te 0.

$$5x^{2}-70x=-245$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-70x}{5}=-\frac{245}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\left(-\frac{70}{5}\right)x=-\frac{245}{5}$$

Whakawehe -70 ki te 5.

$$x^{2}-14x=-\frac{245}{5}$$

Whakawehe -245 ki te 5.

$$x^{2}-14x=-49$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-49+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-49+49$$

Tāpiri -49 ki te 49.

$$x^{2}-14x+49=0$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=0$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-7=0 x-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=7 x=7$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=7$$