Whakaoti i te 9:6x^2-x=15

$$9 : 6 x ^ { 2 } - x = 15$$

Whakaoti mō x

$$x = \frac{\sqrt{91} + 1}{3} \approx 3.513130671$$

$$x=\frac{1-\sqrt{91}}{3}\approx -2.846464005$$

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.

$$\frac{3}{2}x^{2}-x=15$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$\frac{3}{2}x^{2}-x-15=15-15$$

Mā te tango i te 15 i a ia ake anō ka toe ko te 0.

$$\frac{3}{2}x^{2}-x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{3}{2} mō a, -1 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{3}{2}\left(-15\right)}}{2\times \frac{3}{2}}$$

Whakareatia -4 ki te \frac{3}{2}.

$$x=\frac{-\left(-1\right)±\sqrt{1-6\left(-15\right)}}{2\times \frac{3}{2}}$$

Whakareatia -6 ki te -15.

$$x=\frac{-\left(-1\right)±\sqrt{1+90}}{2\times \frac{3}{2}}$$

Tāpiri 1 ki te 90.

$$x=\frac{-\left(-1\right)±\sqrt{91}}{2\times \frac{3}{2}}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{91}}{2\times \frac{3}{2}}$$

Whakareatia 2 ki te \frac{3}{2}.

$$x=\frac{1±\sqrt{91}}{3}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{91}}{3} ina he tāpiri te ±. Tāpiri 1 ki te \sqrt{91}.

$$x=\frac{\sqrt{91}+1}{3}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{91}}{3} ina he tango te ±. Tango \sqrt{91} mai i 1.

$$x=\frac{1-\sqrt{91}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{91}+1}{3} x=\frac{1-\sqrt{91}}{3}$$

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.

$$\frac{3}{2}x^{2}-x=15$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{3}{2}x^{2}-x}{\frac{3}{2}}=\frac{15}{\frac{3}{2}}$$

Mā te whakawehe ki te \frac{3}{2} ka wetekia te whakareanga ki te \frac{3}{2}.

$$x^{2}+\left(-\frac{1}{\frac{3}{2}}\right)x=\frac{15}{\frac{3}{2}}$$

Whakawehe -1 ki te \frac{3}{2} mā te whakarea -1 ki te tau huripoki o \frac{3}{2}.

$$x^{2}-\frac{2}{3}x=\frac{15}{\frac{3}{2}}$$

Whakawehe 15 ki te \frac{3}{2} mā te whakarea 15 ki te tau huripoki o \frac{3}{2}.

$$x^{2}-\frac{2}{3}x=10$$

Whakawehea te -\frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{3}. Nā, tāpiria te pūrua o te -\frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=10+\left(-\frac{1}{3}\right)^{2}$$

Pūruatia -\frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=10+\frac{1}{9}$$

Tāpiri 10 ki te \frac{1}{9}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{91}{9}$$

Tauwehea x^{2}-\frac{2}{3}x+\frac{1}{9}. 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}.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{91}{9}$$

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

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{91}{9}}$$

Whakarūnātia.

$$x-\frac{1}{3}=\frac{\sqrt{91}}{3} x-\frac{1}{3}=-\frac{\sqrt{91}}{3}$$

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{91}+1}{3} x=\frac{1-\sqrt{91}}{3}$$