Whakaoti i te 4x^2+14x-27=0

$$4 x ^ { 2 } + 14 x - 27 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{157} - 7}{4} \approx 1.382491022$$

$$x=\frac{-\sqrt{157}-7}{4}\approx -4.882491022$$

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.

$$4x^{2}+14x-27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 14 mō b, me -27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 4\left(-27\right)}}{2\times 4}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 4\left(-27\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-14±\sqrt{196-16\left(-27\right)}}{2\times 4}$$

Whakareatia -16 ki te -27.

$$x=\frac{-14±\sqrt{196+432}}{2\times 4}$$

Tāpiri 196 ki te 432.

$$x=\frac{-14±\sqrt{628}}{2\times 4}$$

Tuhia te pūtakerua o te 628.

$$x=\frac{-14±2\sqrt{157}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-14±2\sqrt{157}}{8}$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{157}}{8} ina he tāpiri te ±. Tāpiri -14 ki te 2\sqrt{157}.

$$x=\frac{2\sqrt{157}-14}{8}$$

Whakawehe -14+2\sqrt{157} ki te 8.

$$x=\frac{\sqrt{157}-7}{4}$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{157}}{8} ina he tango te ±. Tango 2\sqrt{157} mai i -14.

$$x=\frac{-2\sqrt{157}-14}{8}$$

Whakawehe -14-2\sqrt{157} ki te 8.

$$x=\frac{-\sqrt{157}-7}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{157}-7}{4} x=\frac{-\sqrt{157}-7}{4}$$

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.

$$4x^{2}+14x-27=0$$

Me tāpiri 27 ki ngā taha e rua o te whārite.

$$4x^{2}+14x-27-\left(-27\right)=-\left(-27\right)$$

Mā te tango i te -27 i a ia ake anō ka toe ko te 0.

$$4x^{2}+14x=-\left(-27\right)$$

Tango -27 mai i 0.

$$4x^{2}+14x=27$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+14x}{4}=\frac{27}{4}$$

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

$$x^{2}+\frac{14}{4}x=\frac{27}{4}$$

Whakahekea te hautanga \frac{14}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{7}{2}x=\frac{27}{4}$$

Whakawehea te \frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{4}. Nā, tāpiria te pūrua o te \frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=\frac{27}{4}+\left(\frac{7}{4}\right)^{2}$$

Pūruatia \frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{27}{4}+\frac{49}{16}$$

Tāpiri \frac{27}{4} ki te \frac{49}{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.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{157}{16}$$

Tauwehea x^{2}+\frac{7}{2}x+\frac{49}{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}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{157}{16}$$

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

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{157}{16}}$$

Whakarūnātia.

$$x+\frac{7}{4}=\frac{\sqrt{157}}{4} x+\frac{7}{4}=-\frac{\sqrt{157}}{4}$$

Me tango \frac{7}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{157}-7}{4} x=\frac{-\sqrt{157}-7}{4}$$