Whakaoti i te 28x^2-8x-48=0

$$28 x ^ { 2 } - 8 x - 48 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{85} + 1}{7} \approx 1.459934922$$

$$x=\frac{1-\sqrt{85}}{7}\approx -1.174220637$$

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.

$$28x^{2}-8x-48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 28 mō a, -8 mō b, me -48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 28\left(-48\right)}}{2\times 28}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 28\left(-48\right)}}{2\times 28}$$

Whakareatia -4 ki te 28.

$$x=\frac{-\left(-8\right)±\sqrt{64-112\left(-48\right)}}{2\times 28}$$

Whakareatia -112 ki te -48.

$$x=\frac{-\left(-8\right)±\sqrt{64+5376}}{2\times 28}$$

Tāpiri 64 ki te 5376.

$$x=\frac{-\left(-8\right)±\sqrt{5440}}{2\times 28}$$

Tuhia te pūtakerua o te 5440.

$$x=\frac{-\left(-8\right)±8\sqrt{85}}{2\times 28}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±8\sqrt{85}}{2\times 28}$$

Whakareatia 2 ki te 28.

$$x=\frac{8±8\sqrt{85}}{56}$$

Nā, me whakaoti te whārite x=\frac{8±8\sqrt{85}}{56} ina he tāpiri te ±. Tāpiri 8 ki te 8\sqrt{85}.

$$x=\frac{8\sqrt{85}+8}{56}$$

Whakawehe 8+8\sqrt{85} ki te 56.

$$x=\frac{\sqrt{85}+1}{7}$$

Nā, me whakaoti te whārite x=\frac{8±8\sqrt{85}}{56} ina he tango te ±. Tango 8\sqrt{85} mai i 8.

$$x=\frac{8-8\sqrt{85}}{56}$$

Whakawehe 8-8\sqrt{85} ki te 56.

$$x=\frac{1-\sqrt{85}}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{85}+1}{7} x=\frac{1-\sqrt{85}}{7}$$

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.

$$28x^{2}-8x-48=0$$

Me tāpiri 48 ki ngā taha e rua o te whārite.

$$28x^{2}-8x-48-\left(-48\right)=-\left(-48\right)$$

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

$$28x^{2}-8x=-\left(-48\right)$$

Tango -48 mai i 0.

$$28x^{2}-8x=48$$

Whakawehea ngā taha e rua ki te 28.

$$\frac{28x^{2}-8x}{28}=\frac{48}{28}$$

Mā te whakawehe ki te 28 ka wetekia te whakareanga ki te 28.

$$x^{2}+\left(-\frac{8}{28}\right)x=\frac{48}{28}$$

Whakahekea te hautanga \frac{-8}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{2}{7}x=\frac{48}{28}$$

Whakahekea te hautanga \frac{48}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{2}{7}x=\frac{12}{7}$$

Whakawehea te -\frac{2}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{7}. Nā, tāpiria te pūrua o te -\frac{1}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2}{7}x+\left(-\frac{1}{7}\right)^{2}=\frac{12}{7}+\left(-\frac{1}{7}\right)^{2}$$

Pūruatia -\frac{1}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{12}{7}+\frac{1}{49}$$

Tāpiri \frac{12}{7} ki te \frac{1}{49} 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.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{85}{49}$$

Tauwehea x^{2}-\frac{2}{7}x+\frac{1}{49}. 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}.

$$\left(x-\frac{1}{7}\right)^{2}=\frac{85}{49}$$

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

$$\sqrt{\left(x-\frac{1}{7}\right)^{2}}=\sqrt{\frac{85}{49}}$$

Whakarūnātia.

$$x-\frac{1}{7}=\frac{\sqrt{85}}{7} x-\frac{1}{7}=-\frac{\sqrt{85}}{7}$$

Me tāpiri \frac{1}{7} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{85}+1}{7} x=\frac{1-\sqrt{85}}{7}$$