Whakaoti i te 37x^2-70x+25=0

$$37 x ^ { 2 } - 70 x + 25 = 0$$

Whakaoti mō x

$$x = \frac{10 \sqrt{3} + 35}{37} \approx 1.414067786$$

$$x=\frac{35-10\sqrt{3}}{37}\approx 0.477824106$$

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.

$$37x^{2}-70x+25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 37 mō a, -70 mō b, me 25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 37\times 25}}{2\times 37}$$

Pūrua -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 37\times 25}}{2\times 37}$$

Whakareatia -4 ki te 37.

$$x=\frac{-\left(-70\right)±\sqrt{4900-148\times 25}}{2\times 37}$$

Whakareatia -148 ki te 25.

$$x=\frac{-\left(-70\right)±\sqrt{4900-3700}}{2\times 37}$$

Tāpiri 4900 ki te -3700.

$$x=\frac{-\left(-70\right)±\sqrt{1200}}{2\times 37}$$

Tuhia te pūtakerua o te 1200.

$$x=\frac{-\left(-70\right)±20\sqrt{3}}{2\times 37}$$

Ko te tauaro o -70 ko 70.

$$x=\frac{70±20\sqrt{3}}{2\times 37}$$

Whakareatia 2 ki te 37.

$$x=\frac{70±20\sqrt{3}}{74}$$

Nā, me whakaoti te whārite x=\frac{70±20\sqrt{3}}{74} ina he tāpiri te ±. Tāpiri 70 ki te 20\sqrt{3}.

$$x=\frac{20\sqrt{3}+70}{74}$$

Whakawehe 70+20\sqrt{3} ki te 74.

$$x=\frac{10\sqrt{3}+35}{37}$$

Nā, me whakaoti te whārite x=\frac{70±20\sqrt{3}}{74} ina he tango te ±. Tango 20\sqrt{3} mai i 70.

$$x=\frac{70-20\sqrt{3}}{74}$$

Whakawehe 70-20\sqrt{3} ki te 74.

$$x=\frac{35-10\sqrt{3}}{37}$$

Kua oti te whārite te whakatau.

$$x=\frac{10\sqrt{3}+35}{37} x=\frac{35-10\sqrt{3}}{37}$$

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.

$$37x^{2}-70x+25=0$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$37x^{2}-70x+25-25=-25$$

Mā te tango i te 25 i a ia ake anō ka toe ko te 0.

$$37x^{2}-70x=-25$$

Whakawehea ngā taha e rua ki te 37.

$$\frac{37x^{2}-70x}{37}=-\frac{25}{37}$$

Mā te whakawehe ki te 37 ka wetekia te whakareanga ki te 37.

$$x^{2}-\frac{70}{37}x=-\frac{25}{37}$$

Whakawehea te -\frac{70}{37}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{35}{37}. Nā, tāpiria te pūrua o te -\frac{35}{37} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{70}{37}x+\left(-\frac{35}{37}\right)^{2}=-\frac{25}{37}+\left(-\frac{35}{37}\right)^{2}$$

Pūruatia -\frac{35}{37} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{70}{37}x+\frac{1225}{1369}=-\frac{25}{37}+\frac{1225}{1369}$$

Tāpiri -\frac{25}{37} ki te \frac{1225}{1369} 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.

$$x^{2}-\frac{70}{37}x+\frac{1225}{1369}=\frac{300}{1369}$$

Tauwehea x^{2}-\frac{70}{37}x+\frac{1225}{1369}. 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}.

$$\left(x-\frac{35}{37}\right)^{2}=\frac{300}{1369}$$

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

$$\sqrt{\left(x-\frac{35}{37}\right)^{2}}=\sqrt{\frac{300}{1369}}$$

Whakarūnātia.

$$x-\frac{35}{37}=\frac{10\sqrt{3}}{37} x-\frac{35}{37}=-\frac{10\sqrt{3}}{37}$$

Me tāpiri \frac{35}{37} ki ngā taha e rua o te whārite.

$$x=\frac{10\sqrt{3}+35}{37} x=\frac{35-10\sqrt{3}}{37}$$