Whakaoti i te -1/12x^2+2/3x+5/3=0

$$- \frac { 1 } { 12 } x ^ { 2 } + \frac { 2 } { 3 } x + \frac { 5 } { 3 } = 0$$

Whakaoti mō x

$$x=-2$$

$$x=10$$

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.

$$-\frac{1}{12}x^{2}+\frac{2}{3}x+\frac{5}{3}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{12} mō a, \frac{2}{3} mō b, me \frac{5}{3} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{2}{3}±\sqrt{\left(\frac{2}{3}\right)^{2}-4\left(-\frac{1}{12}\right)\times \frac{5}{3}}}{2\left(-\frac{1}{12}\right)}$$

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

$$x=\frac{-\frac{2}{3}±\sqrt{\frac{4}{9}-4\left(-\frac{1}{12}\right)\times \frac{5}{3}}}{2\left(-\frac{1}{12}\right)}$$

Whakareatia -4 ki te -\frac{1}{12}.

$$x=\frac{-\frac{2}{3}±\sqrt{\frac{4}{9}+\frac{1}{3}\times \frac{5}{3}}}{2\left(-\frac{1}{12}\right)}$$

Whakareatia \frac{1}{3} ki te \frac{5}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\frac{2}{3}±\sqrt{\frac{4+5}{9}}}{2\left(-\frac{1}{12}\right)}$$

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

$$x=\frac{-\frac{2}{3}±\sqrt{1}}{2\left(-\frac{1}{12}\right)}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-\frac{2}{3}±1}{2\left(-\frac{1}{12}\right)}$$

Whakareatia 2 ki te -\frac{1}{12}.

$$x=\frac{-\frac{2}{3}±1}{-\frac{1}{6}}$$

Nā, me whakaoti te whārite x=\frac{-\frac{2}{3}±1}{-\frac{1}{6}} ina he tāpiri te ±. Tāpiri -\frac{2}{3} ki te 1.

$$x=\frac{\frac{1}{3}}{-\frac{1}{6}}$$

Whakawehe \frac{1}{3} ki te -\frac{1}{6} mā te whakarea \frac{1}{3} ki te tau huripoki o -\frac{1}{6}.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-\frac{2}{3}±1}{-\frac{1}{6}} ina he tango te ±. Tango 1 mai i -\frac{2}{3}.

$$x=-\frac{\frac{5}{3}}{-\frac{1}{6}}$$

Whakawehe -\frac{5}{3} ki te -\frac{1}{6} mā te whakarea -\frac{5}{3} ki te tau huripoki o -\frac{1}{6}.

$$x=10$$

Kua oti te whārite te whakatau.

$$x=-2 x=10$$

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.

$$-\frac{1}{12}x^{2}+\frac{2}{3}x+\frac{5}{3}=0$$

Me tango \frac{5}{3} mai i ngā taha e rua o te whārite.

$$-\frac{1}{12}x^{2}+\frac{2}{3}x+\frac{5}{3}-\frac{5}{3}=-\frac{5}{3}$$

Mā te tango i te \frac{5}{3} i a ia ake anō ka toe ko te 0.

$$-\frac{1}{12}x^{2}+\frac{2}{3}x=-\frac{5}{3}$$

Me whakarea ngā taha e rua ki te -12.

$$\frac{-\frac{1}{12}x^{2}+\frac{2}{3}x}{-\frac{1}{12}}=-\frac{\frac{5}{3}}{-\frac{1}{12}}$$

Mā te whakawehe ki te -\frac{1}{12} ka wetekia te whakareanga ki te -\frac{1}{12}.

$$x^{2}+\frac{\frac{2}{3}}{-\frac{1}{12}}x=-\frac{\frac{5}{3}}{-\frac{1}{12}}$$

Whakawehe \frac{2}{3} ki te -\frac{1}{12} mā te whakarea \frac{2}{3} ki te tau huripoki o -\frac{1}{12}.

$$x^{2}-8x=-\frac{\frac{5}{3}}{-\frac{1}{12}}$$

Whakawehe -\frac{5}{3} ki te -\frac{1}{12} mā te whakarea -\frac{5}{3} ki te tau huripoki o -\frac{1}{12}.

$$x^{2}-8x=20$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=20+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=20+16$$

Tāpiri 20 ki te 16.

$$x^{2}-8x+16=36$$

Tauwehea x^{2}-8x+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}.

$$\left(x-4\right)^{2}=36$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{36}$$

Whakarūnātia.

$$x-4=6 x-4=-6$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=10 x=-2$$