Whakaoti i te 5x^2-32x=48

$$5 { x }^{ 2 } -32x=48$$

Whakaoti mō x

$$x = \frac{4 \sqrt{31} + 16}{5} \approx 7.65421149$$

$$x=\frac{16-4\sqrt{31}}{5}\approx -1.25421149$$

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.

$$5x^{2}-32x=48$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$5x^{2}-32x-48=48-48$$

Mā te tango i te 48 i a ia ake anō ka toe ko te 0.

$$5x^{2}-32x-48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -32 mō b, me -48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 5\left(-48\right)}}{2\times 5}$$

Pūrua -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 5\left(-48\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-32\right)±\sqrt{1024-20\left(-48\right)}}{2\times 5}$$

Whakareatia -20 ki te -48.

$$x=\frac{-\left(-32\right)±\sqrt{1024+960}}{2\times 5}$$

Tāpiri 1024 ki te 960.

$$x=\frac{-\left(-32\right)±\sqrt{1984}}{2\times 5}$$

Tuhia te pūtakerua o te 1984.

$$x=\frac{-\left(-32\right)±8\sqrt{31}}{2\times 5}$$

Ko te tauaro o -32 ko 32.

$$x=\frac{32±8\sqrt{31}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{32±8\sqrt{31}}{10}$$

Nā, me whakaoti te whārite x=\frac{32±8\sqrt{31}}{10} ina he tāpiri te ±. Tāpiri 32 ki te 8\sqrt{31}.

$$x=\frac{8\sqrt{31}+32}{10}$$

Whakawehe 32+8\sqrt{31} ki te 10.

$$x=\frac{4\sqrt{31}+16}{5}$$

Nā, me whakaoti te whārite x=\frac{32±8\sqrt{31}}{10} ina he tango te ±. Tango 8\sqrt{31} mai i 32.

$$x=\frac{32-8\sqrt{31}}{10}$$

Whakawehe 32-8\sqrt{31} ki te 10.

$$x=\frac{16-4\sqrt{31}}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{4\sqrt{31}+16}{5} x=\frac{16-4\sqrt{31}}{5}$$

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.

$$5x^{2}-32x=48$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-32x}{5}=\frac{48}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{32}{5}x=\frac{48}{5}$$

Whakawehea te -\frac{32}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{16}{5}. Nā, tāpiria te pūrua o te -\frac{16}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{32}{5}x+\left(-\frac{16}{5}\right)^{2}=\frac{48}{5}+\left(-\frac{16}{5}\right)^{2}$$

Pūruatia -\frac{16}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{32}{5}x+\frac{256}{25}=\frac{48}{5}+\frac{256}{25}$$

Tāpiri \frac{48}{5} ki te \frac{256}{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.

$$x^{2}-\frac{32}{5}x+\frac{256}{25}=\frac{496}{25}$$

Tauwehea x^{2}-\frac{32}{5}x+\frac{256}{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}.

$$\left(x-\frac{16}{5}\right)^{2}=\frac{496}{25}$$

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

$$\sqrt{\left(x-\frac{16}{5}\right)^{2}}=\sqrt{\frac{496}{25}}$$

Whakarūnātia.

$$x-\frac{16}{5}=\frac{4\sqrt{31}}{5} x-\frac{16}{5}=-\frac{4\sqrt{31}}{5}$$

Me tāpiri \frac{16}{5} ki ngā taha e rua o te whārite.

$$x=\frac{4\sqrt{31}+16}{5} x=\frac{16-4\sqrt{31}}{5}$$