Whakaoti i te 5x^2-48x-48=0

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

Whakaoti mō x

$$x = \frac{4 \sqrt{51} + 24}{5} \approx 10.513142743$$

$$x=\frac{24-4\sqrt{51}}{5}\approx -0.913142743$$

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.

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

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

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

Pūrua -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 5\left(-48\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-48\right)±\sqrt{2304-20\left(-48\right)}}{2\times 5}$$

Whakareatia -20 ki te -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304+960}}{2\times 5}$$

Tāpiri 2304 ki te 960.

$$x=\frac{-\left(-48\right)±\sqrt{3264}}{2\times 5}$$

Tuhia te pūtakerua o te 3264.

$$x=\frac{-\left(-48\right)±8\sqrt{51}}{2\times 5}$$

Ko te tauaro o -48 ko 48.

$$x=\frac{48±8\sqrt{51}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{48±8\sqrt{51}}{10}$$

Nā, me whakaoti te whārite x=\frac{48±8\sqrt{51}}{10} ina he tāpiri te ±. Tāpiri 48 ki te 8\sqrt{51}.

$$x=\frac{8\sqrt{51}+48}{10}$$

Whakawehe 48+8\sqrt{51} ki te 10.

$$x=\frac{4\sqrt{51}+24}{5}$$

Nā, me whakaoti te whārite x=\frac{48±8\sqrt{51}}{10} ina he tango te ±. Tango 8\sqrt{51} mai i 48.

$$x=\frac{48-8\sqrt{51}}{10}$$

Whakawehe 48-8\sqrt{51} ki te 10.

$$x=\frac{24-4\sqrt{51}}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{4\sqrt{51}+24}{5} x=\frac{24-4\sqrt{51}}{5}$$

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.

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

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

$$5x^{2}-48x-48-\left(-48\right)=-\left(-48\right)$$

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

$$5x^{2}-48x=-\left(-48\right)$$

Tango -48 mai i 0.

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

Whakawehea ngā taha e rua ki te 5.

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

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

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

Whakawehea te -\frac{48}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{24}{5}. Nā, tāpiria te pūrua o te -\frac{24}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{48}{5}x+\left(-\frac{24}{5}\right)^{2}=\frac{48}{5}+\left(-\frac{24}{5}\right)^{2}$$

Pūruatia -\frac{24}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=\frac{48}{5}+\frac{576}{25}$$

Tāpiri \frac{48}{5} ki te \frac{576}{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.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=\frac{816}{25}$$

Tauwehea x^{2}-\frac{48}{5}x+\frac{576}{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}.

$$\left(x-\frac{24}{5}\right)^{2}=\frac{816}{25}$$

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

$$\sqrt{\left(x-\frac{24}{5}\right)^{2}}=\sqrt{\frac{816}{25}}$$

Whakarūnātia.

$$x-\frac{24}{5}=\frac{4\sqrt{51}}{5} x-\frac{24}{5}=-\frac{4\sqrt{51}}{5}$$

Me tāpiri \frac{24}{5} ki ngā taha e rua o te whārite.

$$x=\frac{4\sqrt{51}+24}{5} x=\frac{24-4\sqrt{51}}{5}$$