Whakaoti i te 3x^2=9x-5

$$3 x ^ { 2 } = 9 x - 5$$

Whakaoti mō x

$$x=\frac{\sqrt{21}}{6}+\frac{3}{2}\approx 2.263762616$$

$$x=-\frac{\sqrt{21}}{6}+\frac{3}{2}\approx 0.736237384$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 9x mai i ngā taha e rua.

$$3x^{2}-9x=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$3x^{2}-9x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -9 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 3\times 5}}{2\times 3}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 3\times 5}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-9\right)±\sqrt{81-12\times 5}}{2\times 3}$$

Whakareatia -12 ki te 5.

$$x=\frac{-\left(-9\right)±\sqrt{81-60}}{2\times 3}$$

Tāpiri 81 ki te -60.

$$x=\frac{-\left(-9\right)±\sqrt{21}}{2\times 3}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±\sqrt{21}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{9±\sqrt{21}}{6}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{21}}{6} ina he tāpiri te ±. Tāpiri 9 ki te \sqrt{21}.

$$x=\frac{\sqrt{21}+9}{6}$$

Whakawehe 9+\sqrt{21} ki te 6.

$$x=\frac{\sqrt{21}}{6}+\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{21}}{6} ina he tango te ±. Tango \sqrt{21} mai i 9.

$$x=\frac{9-\sqrt{21}}{6}$$

Whakawehe 9-\sqrt{21} ki te 6.

$$x=-\frac{\sqrt{21}}{6}+\frac{3}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{21}}{6}+\frac{3}{2} x=-\frac{\sqrt{21}}{6}+\frac{3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 9x mai i ngā taha e rua.

$$3x^{2}-9x=-5$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-9x}{3}=-\frac{5}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\left(-\frac{9}{3}\right)x=-\frac{5}{3}$$

Whakawehe -9 ki te 3.

$$x^{2}-3x=-\frac{5}{3}$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-\frac{5}{3}+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=-\frac{5}{3}+\frac{9}{4}$$

Tāpiri -\frac{5}{3} ki te \frac{9}{4} 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.

$$x^{2}-3x+\frac{9}{4}=\frac{7}{12}$$

Tauwehea x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{7}{12}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{7}{12}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{\sqrt{21}}{6} x-\frac{3}{2}=-\frac{\sqrt{21}}{6}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{21}}{6}+\frac{3}{2} x=-\frac{\sqrt{21}}{6}+\frac{3}{2}$$