Whakaoti i te 48x^2-52x-26=0

$$48 x ^ { 2 } - 52 x - 26 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{481} + 13}{24} \approx 1.455488008$$

$$x=\frac{13-\sqrt{481}}{24}\approx -0.372154675$$

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.

$$48x^{2}-52x-26=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 48 mō a, -52 mō b, me -26 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 48\left(-26\right)}}{2\times 48}$$

Pūrua -52.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 48\left(-26\right)}}{2\times 48}$$

Whakareatia -4 ki te 48.

$$x=\frac{-\left(-52\right)±\sqrt{2704-192\left(-26\right)}}{2\times 48}$$

Whakareatia -192 ki te -26.

$$x=\frac{-\left(-52\right)±\sqrt{2704+4992}}{2\times 48}$$

Tāpiri 2704 ki te 4992.

$$x=\frac{-\left(-52\right)±\sqrt{7696}}{2\times 48}$$

Tuhia te pūtakerua o te 7696.

$$x=\frac{-\left(-52\right)±4\sqrt{481}}{2\times 48}$$

Ko te tauaro o -52 ko 52.

$$x=\frac{52±4\sqrt{481}}{2\times 48}$$

Whakareatia 2 ki te 48.

$$x=\frac{52±4\sqrt{481}}{96}$$

Nā, me whakaoti te whārite x=\frac{52±4\sqrt{481}}{96} ina he tāpiri te ±. Tāpiri 52 ki te 4\sqrt{481}.

$$x=\frac{4\sqrt{481}+52}{96}$$

Whakawehe 52+4\sqrt{481} ki te 96.

$$x=\frac{\sqrt{481}+13}{24}$$

Nā, me whakaoti te whārite x=\frac{52±4\sqrt{481}}{96} ina he tango te ±. Tango 4\sqrt{481} mai i 52.

$$x=\frac{52-4\sqrt{481}}{96}$$

Whakawehe 52-4\sqrt{481} ki te 96.

$$x=\frac{13-\sqrt{481}}{24}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{481}+13}{24} x=\frac{13-\sqrt{481}}{24}$$

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.

$$48x^{2}-52x-26=0$$

Me tāpiri 26 ki ngā taha e rua o te whārite.

$$48x^{2}-52x-26-\left(-26\right)=-\left(-26\right)$$

Mā te tango i te -26 i a ia ake anō ka toe ko te 0.

$$48x^{2}-52x=-\left(-26\right)$$

Tango -26 mai i 0.

$$48x^{2}-52x=26$$

Whakawehea ngā taha e rua ki te 48.

$$\frac{48x^{2}-52x}{48}=\frac{26}{48}$$

Mā te whakawehe ki te 48 ka wetekia te whakareanga ki te 48.

$$x^{2}+\left(-\frac{52}{48}\right)x=\frac{26}{48}$$

Whakahekea te hautanga \frac{-52}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{13}{12}x=\frac{26}{48}$$

Whakahekea te hautanga \frac{26}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{13}{12}x=\frac{13}{24}$$

Whakawehea te -\frac{13}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{24}. Nā, tāpiria te pūrua o te -\frac{13}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{12}x+\left(-\frac{13}{24}\right)^{2}=\frac{13}{24}+\left(-\frac{13}{24}\right)^{2}$$

Pūruatia -\frac{13}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{12}x+\frac{169}{576}=\frac{13}{24}+\frac{169}{576}$$

Tāpiri \frac{13}{24} ki te \frac{169}{576} 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.

$$x^{2}-\frac{13}{12}x+\frac{169}{576}=\frac{481}{576}$$

Tauwehea te x^{2}-\frac{13}{12}x+\frac{169}{576}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{24}\right)^{2}=\frac{481}{576}$$

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

$$\sqrt{\left(x-\frac{13}{24}\right)^{2}}=\sqrt{\frac{481}{576}}$$

Whakarūnātia.

$$x-\frac{13}{24}=\frac{\sqrt{481}}{24} x-\frac{13}{24}=-\frac{\sqrt{481}}{24}$$

Me tāpiri \frac{13}{24} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{481}+13}{24} x=\frac{13-\sqrt{481}}{24}$$