Whakaoti i te 20x^2-1584-392x=0

$$20 x ^ { 2 } - 1584 - 392 x = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{4381} + 49}{5} \approx 23.037824595$$

$$x=\frac{49-\sqrt{4381}}{5}\approx -3.437824595$$

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.

$$20x^{2}-392x-1584=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 20 mō a, -392 mō b, me -1584 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-392\right)±\sqrt{\left(-392\right)^{2}-4\times 20\left(-1584\right)}}{2\times 20}$$

Pūrua -392.

$$x=\frac{-\left(-392\right)±\sqrt{153664-4\times 20\left(-1584\right)}}{2\times 20}$$

Whakareatia -4 ki te 20.

$$x=\frac{-\left(-392\right)±\sqrt{153664-80\left(-1584\right)}}{2\times 20}$$

Whakareatia -80 ki te -1584.

$$x=\frac{-\left(-392\right)±\sqrt{153664+126720}}{2\times 20}$$

Tāpiri 153664 ki te 126720.

$$x=\frac{-\left(-392\right)±\sqrt{280384}}{2\times 20}$$

Tuhia te pūtakerua o te 280384.

$$x=\frac{-\left(-392\right)±8\sqrt{4381}}{2\times 20}$$

Ko te tauaro o -392 ko 392.

$$x=\frac{392±8\sqrt{4381}}{2\times 20}$$

Whakareatia 2 ki te 20.

$$x=\frac{392±8\sqrt{4381}}{40}$$

Nā, me whakaoti te whārite x=\frac{392±8\sqrt{4381}}{40} ina he tāpiri te ±. Tāpiri 392 ki te 8\sqrt{4381}.

$$x=\frac{8\sqrt{4381}+392}{40}$$

Whakawehe 392+8\sqrt{4381} ki te 40.

$$x=\frac{\sqrt{4381}+49}{5}$$

Nā, me whakaoti te whārite x=\frac{392±8\sqrt{4381}}{40} ina he tango te ±. Tango 8\sqrt{4381} mai i 392.

$$x=\frac{392-8\sqrt{4381}}{40}$$

Whakawehe 392-8\sqrt{4381} ki te 40.

$$x=\frac{49-\sqrt{4381}}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{4381}+49}{5} x=\frac{49-\sqrt{4381}}{5}$$

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.

$$20x^{2}-392x-1584=0$$

Me tāpiri 1584 ki ngā taha e rua o te whārite.

$$20x^{2}-392x-1584-\left(-1584\right)=-\left(-1584\right)$$

Mā te tango i te -1584 i a ia ake anō ka toe ko te 0.

$$20x^{2}-392x=-\left(-1584\right)$$

Tango -1584 mai i 0.

$$20x^{2}-392x=1584$$

Whakawehea ngā taha e rua ki te 20.

$$\frac{20x^{2}-392x}{20}=\frac{1584}{20}$$

Mā te whakawehe ki te 20 ka wetekia te whakareanga ki te 20.

$$x^{2}+\left(-\frac{392}{20}\right)x=\frac{1584}{20}$$

Whakahekea te hautanga \frac{-392}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{98}{5}x=\frac{1584}{20}$$

Whakahekea te hautanga \frac{1584}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{98}{5}x=\frac{396}{5}$$

Whakawehea te -\frac{98}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{49}{5}. Nā, tāpiria te pūrua o te -\frac{49}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{98}{5}x+\left(-\frac{49}{5}\right)^{2}=\frac{396}{5}+\left(-\frac{49}{5}\right)^{2}$$

Pūruatia -\frac{49}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{98}{5}x+\frac{2401}{25}=\frac{396}{5}+\frac{2401}{25}$$

Tāpiri \frac{396}{5} ki te \frac{2401}{25} 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.

$$x^{2}-\frac{98}{5}x+\frac{2401}{25}=\frac{4381}{25}$$

Tauwehea x^{2}-\frac{98}{5}x+\frac{2401}{25}. 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}.

$$\left(x-\frac{49}{5}\right)^{2}=\frac{4381}{25}$$

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

$$\sqrt{\left(x-\frac{49}{5}\right)^{2}}=\sqrt{\frac{4381}{25}}$$

Whakarūnātia.

$$x-\frac{49}{5}=\frac{\sqrt{4381}}{5} x-\frac{49}{5}=-\frac{\sqrt{4381}}{5}$$

Me tāpiri \frac{49}{5} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{4381}+49}{5} x=\frac{49-\sqrt{4381}}{5}$$