Whakaoti i te 4x^2-75x+50=0

$$4 x ^ { 2 } - 75 x + 50 = 0$$

Whakaoti mō x

$$x = \frac{5 \sqrt{193} + 75}{8} \approx 18.057777493$$

$$x=\frac{75-5\sqrt{193}}{8}\approx 0.692222507$$

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.

$$4x^{2}-75x+50=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -75 mō b, me 50 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-75\right)±\sqrt{\left(-75\right)^{2}-4\times 4\times 50}}{2\times 4}$$

Pūrua -75.

$$x=\frac{-\left(-75\right)±\sqrt{5625-4\times 4\times 50}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-75\right)±\sqrt{5625-16\times 50}}{2\times 4}$$

Whakareatia -16 ki te 50.

$$x=\frac{-\left(-75\right)±\sqrt{5625-800}}{2\times 4}$$

Tāpiri 5625 ki te -800.

$$x=\frac{-\left(-75\right)±\sqrt{4825}}{2\times 4}$$

Tuhia te pūtakerua o te 4825.

$$x=\frac{-\left(-75\right)±5\sqrt{193}}{2\times 4}$$

Ko te tauaro o -75 ko 75.

$$x=\frac{75±5\sqrt{193}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{75±5\sqrt{193}}{8}$$

Nā, me whakaoti te whārite x=\frac{75±5\sqrt{193}}{8} ina he tāpiri te ±. Tāpiri 75 ki te 5\sqrt{193}.

$$x=\frac{5\sqrt{193}+75}{8}$$

Nā, me whakaoti te whārite x=\frac{75±5\sqrt{193}}{8} ina he tango te ±. Tango 5\sqrt{193} mai i 75.

$$x=\frac{75-5\sqrt{193}}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{5\sqrt{193}+75}{8} x=\frac{75-5\sqrt{193}}{8}$$

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.

$$4x^{2}-75x+50=0$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$4x^{2}-75x+50-50=-50$$

Mā te tango i te 50 i a ia ake anō ka toe ko te 0.

$$4x^{2}-75x=-50$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-75x}{4}=-\frac{50}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}-\frac{75}{4}x=-\frac{50}{4}$$

Whakahekea te hautanga \frac{-50}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{75}{4}x=-\frac{25}{2}$$

Whakawehea te -\frac{75}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{75}{8}. Nā, tāpiria te pūrua o te -\frac{75}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{75}{4}x+\left(-\frac{75}{8}\right)^{2}=-\frac{25}{2}+\left(-\frac{75}{8}\right)^{2}$$

Pūruatia -\frac{75}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{75}{4}x+\frac{5625}{64}=-\frac{25}{2}+\frac{5625}{64}$$

Tāpiri -\frac{25}{2} ki te \frac{5625}{64} 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.

$$x^{2}-\frac{75}{4}x+\frac{5625}{64}=\frac{4825}{64}$$

Tauwehea x^{2}-\frac{75}{4}x+\frac{5625}{64}. 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}.

$$\left(x-\frac{75}{8}\right)^{2}=\frac{4825}{64}$$

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

$$\sqrt{\left(x-\frac{75}{8}\right)^{2}}=\sqrt{\frac{4825}{64}}$$

Whakarūnātia.

$$x-\frac{75}{8}=\frac{5\sqrt{193}}{8} x-\frac{75}{8}=-\frac{5\sqrt{193}}{8}$$

Me tāpiri \frac{75}{8} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{193}+75}{8} x=\frac{75-5\sqrt{193}}{8}$$