Whakaoti i te 27x^2+59x-21=0

$$27 { x }^{ 2 } +59x-21=0$$

Whakaoti mō x

$$x=\frac{\sqrt{5749}-59}{54}\approx 0.311521488$$

$$x=\frac{-\sqrt{5749}-59}{54}\approx -2.496706673$$

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.

$$27x^{2}+59x-21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 27 mō a, 59 mō b, me -21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-59±\sqrt{59^{2}-4\times 27\left(-21\right)}}{2\times 27}$$

Pūrua 59.

$$x=\frac{-59±\sqrt{3481-4\times 27\left(-21\right)}}{2\times 27}$$

Whakareatia -4 ki te 27.

$$x=\frac{-59±\sqrt{3481-108\left(-21\right)}}{2\times 27}$$

Whakareatia -108 ki te -21.

$$x=\frac{-59±\sqrt{3481+2268}}{2\times 27}$$

Tāpiri 3481 ki te 2268.

$$x=\frac{-59±\sqrt{5749}}{2\times 27}$$

Whakareatia 2 ki te 27.

$$x=\frac{-59±\sqrt{5749}}{54}$$

Nā, me whakaoti te whārite x=\frac{-59±\sqrt{5749}}{54} ina he tāpiri te ±. Tāpiri -59 ki te \sqrt{5749}.

$$x=\frac{\sqrt{5749}-59}{54}$$

Nā, me whakaoti te whārite x=\frac{-59±\sqrt{5749}}{54} ina he tango te ±. Tango \sqrt{5749} mai i -59.

$$x=\frac{-\sqrt{5749}-59}{54}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{5749}-59}{54} x=\frac{-\sqrt{5749}-59}{54}$$

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.

$$27x^{2}+59x-21=0$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$27x^{2}+59x-21-\left(-21\right)=-\left(-21\right)$$

Mā te tango i te -21 i a ia ake anō ka toe ko te 0.

$$27x^{2}+59x=-\left(-21\right)$$

Tango -21 mai i 0.

$$27x^{2}+59x=21$$

Whakawehea ngā taha e rua ki te 27.

$$\frac{27x^{2}+59x}{27}=\frac{21}{27}$$

Mā te whakawehe ki te 27 ka wetekia te whakareanga ki te 27.

$$x^{2}+\frac{59}{27}x=\frac{21}{27}$$

Whakahekea te hautanga \frac{21}{27} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{59}{27}x=\frac{7}{9}$$

Whakawehea te \frac{59}{27}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{59}{54}. Nā, tāpiria te pūrua o te \frac{59}{54} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{59}{27}x+\left(\frac{59}{54}\right)^{2}=\frac{7}{9}+\left(\frac{59}{54}\right)^{2}$$

Pūruatia \frac{59}{54} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{59}{27}x+\frac{3481}{2916}=\frac{7}{9}+\frac{3481}{2916}$$

Tāpiri \frac{7}{9} ki te \frac{3481}{2916} 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.

$$x^{2}+\frac{59}{27}x+\frac{3481}{2916}=\frac{5749}{2916}$$

Tauwehea x^{2}+\frac{59}{27}x+\frac{3481}{2916}. 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}.

$$\left(x+\frac{59}{54}\right)^{2}=\frac{5749}{2916}$$

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

$$\sqrt{\left(x+\frac{59}{54}\right)^{2}}=\sqrt{\frac{5749}{2916}}$$

Whakarūnātia.

$$x+\frac{59}{54}=\frac{\sqrt{5749}}{54} x+\frac{59}{54}=-\frac{\sqrt{5749}}{54}$$

Me tango \frac{59}{54} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{5749}-59}{54} x=\frac{-\sqrt{5749}-59}{54}$$