Whakaoti i te 14x+8x^2=970

$$14x+8 { x }^{ 2 } =970$$

Whakaoti mō x

$$x = \frac{\sqrt{7809} - 7}{8} \approx 10.171068305$$

$$x=\frac{-\sqrt{7809}-7}{8}\approx -11.921068305$$

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.

$$8x^{2}+14x=970$$

Me tango 970 mai i ngā taha e rua o te whārite.

$$8x^{2}+14x-970=970-970$$

Mā te tango i te 970 i a ia ake anō ka toe ko te 0.

$$8x^{2}+14x-970=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, 14 mō b, me -970 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 8\left(-970\right)}}{2\times 8}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 8\left(-970\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-14±\sqrt{196-32\left(-970\right)}}{2\times 8}$$

Whakareatia -32 ki te -970.

$$x=\frac{-14±\sqrt{196+31040}}{2\times 8}$$

Tāpiri 196 ki te 31040.

$$x=\frac{-14±\sqrt{31236}}{2\times 8}$$

Tuhia te pūtakerua o te 31236.

$$x=\frac{-14±2\sqrt{7809}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-14±2\sqrt{7809}}{16}$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{7809}}{16} ina he tāpiri te ±. Tāpiri -14 ki te 2\sqrt{7809}.

$$x=\frac{2\sqrt{7809}-14}{16}$$

Whakawehe -14+2\sqrt{7809} ki te 16.

$$x=\frac{\sqrt{7809}-7}{8}$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{7809}}{16} ina he tango te ±. Tango 2\sqrt{7809} mai i -14.

$$x=\frac{-2\sqrt{7809}-14}{16}$$

Whakawehe -14-2\sqrt{7809} ki te 16.

$$x=\frac{-\sqrt{7809}-7}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{7809}-7}{8} x=\frac{-\sqrt{7809}-7}{8}$$

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.

$$8x^{2}+14x=970$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}+14x}{8}=\frac{970}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\frac{14}{8}x=\frac{970}{8}$$

Whakahekea te hautanga \frac{14}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{7}{4}x=\frac{970}{8}$$

Whakahekea te hautanga \frac{970}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{7}{4}x=\frac{485}{4}$$

Whakawehea te \frac{7}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{8}. Nā, tāpiria te pūrua o te \frac{7}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\frac{485}{4}+\left(\frac{7}{8}\right)^{2}$$

Pūruatia \frac{7}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{485}{4}+\frac{49}{64}$$

Tāpiri \frac{485}{4} ki te \frac{49}{64} 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.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{7809}{64}$$

Tauwehea x^{2}+\frac{7}{4}x+\frac{49}{64}. 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}.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{7809}{64}$$

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

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{7809}{64}}$$

Whakarūnātia.

$$x+\frac{7}{8}=\frac{\sqrt{7809}}{8} x+\frac{7}{8}=-\frac{\sqrt{7809}}{8}$$

Me tango \frac{7}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{7809}-7}{8} x=\frac{-\sqrt{7809}-7}{8}$$