Whakaoti i te 15x^2+44x-46=0

$$15 { x }^{ 2 } +44x-46=0$$

Whakaoti mō x

$$x=\frac{\sqrt{1174}-22}{15}\approx 0.817578893$$

$$x=\frac{-\sqrt{1174}-22}{15}\approx -3.750912227$$

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.

$$15x^{2}+44x-46=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, 44 mō b, me -46 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-44±\sqrt{44^{2}-4\times 15\left(-46\right)}}{2\times 15}$$

Pūrua 44.

$$x=\frac{-44±\sqrt{1936-4\times 15\left(-46\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-44±\sqrt{1936-60\left(-46\right)}}{2\times 15}$$

Whakareatia -60 ki te -46.

$$x=\frac{-44±\sqrt{1936+2760}}{2\times 15}$$

Tāpiri 1936 ki te 2760.

$$x=\frac{-44±\sqrt{4696}}{2\times 15}$$

Tuhia te pūtakerua o te 4696.

$$x=\frac{-44±2\sqrt{1174}}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{-44±2\sqrt{1174}}{30}$$

Nā, me whakaoti te whārite x=\frac{-44±2\sqrt{1174}}{30} ina he tāpiri te ±. Tāpiri -44 ki te 2\sqrt{1174}.

$$x=\frac{2\sqrt{1174}-44}{30}$$

Whakawehe -44+2\sqrt{1174} ki te 30.

$$x=\frac{\sqrt{1174}-22}{15}$$

Nā, me whakaoti te whārite x=\frac{-44±2\sqrt{1174}}{30} ina he tango te ±. Tango 2\sqrt{1174} mai i -44.

$$x=\frac{-2\sqrt{1174}-44}{30}$$

Whakawehe -44-2\sqrt{1174} ki te 30.

$$x=\frac{-\sqrt{1174}-22}{15}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{1174}-22}{15} x=\frac{-\sqrt{1174}-22}{15}$$

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.

$$15x^{2}+44x-46=0$$

Me tāpiri 46 ki ngā taha e rua o te whārite.

$$15x^{2}+44x-46-\left(-46\right)=-\left(-46\right)$$

Mā te tango i te -46 i a ia ake anō ka toe ko te 0.

$$15x^{2}+44x=-\left(-46\right)$$

Tango -46 mai i 0.

$$15x^{2}+44x=46$$

Whakawehea ngā taha e rua ki te 15.

$$\frac{15x^{2}+44x}{15}=\frac{46}{15}$$

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

$$x^{2}+\frac{44}{15}x=\frac{46}{15}$$

Whakawehea te \frac{44}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{22}{15}. Nā, tāpiria te pūrua o te \frac{22}{15} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{44}{15}x+\left(\frac{22}{15}\right)^{2}=\frac{46}{15}+\left(\frac{22}{15}\right)^{2}$$

Pūruatia \frac{22}{15} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{44}{15}x+\frac{484}{225}=\frac{46}{15}+\frac{484}{225}$$

Tāpiri \frac{46}{15} ki te \frac{484}{225} 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.

$$x^{2}+\frac{44}{15}x+\frac{484}{225}=\frac{1174}{225}$$

Tauwehea x^{2}+\frac{44}{15}x+\frac{484}{225}. 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}.

$$\left(x+\frac{22}{15}\right)^{2}=\frac{1174}{225}$$

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

$$\sqrt{\left(x+\frac{22}{15}\right)^{2}}=\sqrt{\frac{1174}{225}}$$

Whakarūnātia.

$$x+\frac{22}{15}=\frac{\sqrt{1174}}{15} x+\frac{22}{15}=-\frac{\sqrt{1174}}{15}$$

Me tango \frac{22}{15} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1174}-22}{15} x=\frac{-\sqrt{1174}-22}{15}$$