Whakaoti i te 6x^2+5/3x-21=0

$$6 x ^ { 2 } + \frac { 5 } { 3 } x - 21 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{4561} - 5}{36} \approx 1.737088223$$

$$x=\frac{-\sqrt{4561}-5}{36}\approx -2.014866001$$

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.

$$6x^{2}+\frac{5}{3}x-21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, \frac{5}{3} mō b, me -21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{5}{3}±\sqrt{\left(\frac{5}{3}\right)^{2}-4\times 6\left(-21\right)}}{2\times 6}$$

Pūruatia \frac{5}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}-4\times 6\left(-21\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}-24\left(-21\right)}}{2\times 6}$$

Whakareatia -24 ki te -21.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}+504}}{2\times 6}$$

Tāpiri \frac{25}{9} ki te 504.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{4561}{9}}}{2\times 6}$$

Tuhia te pūtakerua o te \frac{4561}{9}.

$$x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{12}$$

Nā, me whakaoti te whārite x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{12} ina he tāpiri te ±. Tāpiri -\frac{5}{3} ki te \frac{\sqrt{4561}}{3}.

$$x=\frac{\sqrt{4561}-5}{3\times 12}$$

Whakawehe \frac{-5+\sqrt{4561}}{3} ki te 12.

$$x=\frac{\sqrt{4561}-5}{36}$$

Nā, me whakaoti te whārite x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{12} ina he tango te ±. Tango \frac{\sqrt{4561}}{3} mai i -\frac{5}{3}.

$$x=\frac{-\sqrt{4561}-5}{3\times 12}$$

Whakawehe \frac{-5-\sqrt{4561}}{3} ki te 12.

$$x=\frac{-\sqrt{4561}-5}{36}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{4561}-5}{36} x=\frac{-\sqrt{4561}-5}{36}$$

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.

$$6x^{2}+\frac{5}{3}x-21=0$$

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

$$6x^{2}+\frac{5}{3}x-21-\left(-21\right)=-\left(-21\right)$$

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

$$6x^{2}+\frac{5}{3}x=-\left(-21\right)$$

Tango -21 mai i 0.

$$6x^{2}+\frac{5}{3}x=21$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+\frac{5}{3}x}{6}=\frac{21}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{\frac{5}{3}}{6}x=\frac{21}{6}$$

Whakawehe \frac{5}{3} ki te 6.

$$x^{2}+\frac{5}{18}x=\frac{21}{6}$$

Whakahekea te hautanga \frac{21}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{5}{18}x=\frac{7}{2}$$

Whakawehea te \frac{5}{18}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{36}. Nā, tāpiria te pūrua o te \frac{5}{36} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{18}x+\left(\frac{5}{36}\right)^{2}=\frac{7}{2}+\left(\frac{5}{36}\right)^{2}$$

Pūruatia \frac{5}{36} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{18}x+\frac{25}{1296}=\frac{7}{2}+\frac{25}{1296}$$

Tāpiri \frac{7}{2} ki te \frac{25}{1296} 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.

$$x^{2}+\frac{5}{18}x+\frac{25}{1296}=\frac{4561}{1296}$$

Tauwehea x^{2}+\frac{5}{18}x+\frac{25}{1296}. 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}.

$$\left(x+\frac{5}{36}\right)^{2}=\frac{4561}{1296}$$

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

$$\sqrt{\left(x+\frac{5}{36}\right)^{2}}=\sqrt{\frac{4561}{1296}}$$

Whakarūnātia.

$$x+\frac{5}{36}=\frac{\sqrt{4561}}{36} x+\frac{5}{36}=-\frac{\sqrt{4561}}{36}$$

Me tango \frac{5}{36} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{4561}-5}{36} x=\frac{-\sqrt{4561}-5}{36}$$