Whakaoti i te 4x^2-18x+5=0

$$4 x ^ { 2 } - 18 x + 5 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{61} + 9}{4} \approx 4.202562419$$

$$x=\frac{9-\sqrt{61}}{4}\approx 0.297437581$$

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.

$$4x^{2}-18x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -18 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 4\times 5}}{2\times 4}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 4\times 5}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-18\right)±\sqrt{324-16\times 5}}{2\times 4}$$

Whakareatia -16 ki te 5.

$$x=\frac{-\left(-18\right)±\sqrt{324-80}}{2\times 4}$$

Tāpiri 324 ki te -80.

$$x=\frac{-\left(-18\right)±\sqrt{244}}{2\times 4}$$

Tuhia te pūtakerua o te 244.

$$x=\frac{-\left(-18\right)±2\sqrt{61}}{2\times 4}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±2\sqrt{61}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{18±2\sqrt{61}}{8}$$

Nā, me whakaoti te whārite x=\frac{18±2\sqrt{61}}{8} ina he tāpiri te ±. Tāpiri 18 ki te 2\sqrt{61}.

$$x=\frac{2\sqrt{61}+18}{8}$$

Whakawehe 18+2\sqrt{61} ki te 8.

$$x=\frac{\sqrt{61}+9}{4}$$

Nā, me whakaoti te whārite x=\frac{18±2\sqrt{61}}{8} ina he tango te ±. Tango 2\sqrt{61} mai i 18.

$$x=\frac{18-2\sqrt{61}}{8}$$

Whakawehe 18-2\sqrt{61} ki te 8.

$$x=\frac{9-\sqrt{61}}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{61}+9}{4} x=\frac{9-\sqrt{61}}{4}$$

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.

$$4x^{2}-18x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$4x^{2}-18x+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$4x^{2}-18x=-5$$

Whakawehea ngā taha e rua ki te 4.

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

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

$$x^{2}+\left(-\frac{18}{4}\right)x=-\frac{5}{4}$$

Whakahekea te hautanga \frac{-18}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{9}{2}x=-\frac{5}{4}$$

Whakawehea te -\frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{4}. Nā, tāpiria te pūrua o te -\frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{5}{4}+\frac{81}{16}$$

Tāpiri -\frac{5}{4} ki te \frac{81}{16} 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.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{61}{16}$$

Tauwehea x^{2}-\frac{9}{2}x+\frac{81}{16}. 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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{61}{16}$$

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

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{61}{16}}$$

Whakarūnātia.

$$x-\frac{9}{4}=\frac{\sqrt{61}}{4} x-\frac{9}{4}=-\frac{\sqrt{61}}{4}$$

Me tāpiri \frac{9}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{61}+9}{4} x=\frac{9-\sqrt{61}}{4}$$