Whakaoti i te 5x^2-12x-7=0

$$5 { x }^{ 2 } -12x-7=0$$

Whakaoti mō x

$$x = \frac{\sqrt{71} + 6}{5} \approx 2.885229955$$

$$x=\frac{6-\sqrt{71}}{5}\approx -0.485229955$$

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.

$$5x^{2}-12x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -12 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 5\left(-7\right)}}{2\times 5}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 5\left(-7\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-12\right)±\sqrt{144-20\left(-7\right)}}{2\times 5}$$

Whakareatia -20 ki te -7.

$$x=\frac{-\left(-12\right)±\sqrt{144+140}}{2\times 5}$$

Tāpiri 144 ki te 140.

$$x=\frac{-\left(-12\right)±\sqrt{284}}{2\times 5}$$

Tuhia te pūtakerua o te 284.

$$x=\frac{-\left(-12\right)±2\sqrt{71}}{2\times 5}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±2\sqrt{71}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{12±2\sqrt{71}}{10}$$

Nā, me whakaoti te whārite x=\frac{12±2\sqrt{71}}{10} ina he tāpiri te ±. Tāpiri 12 ki te 2\sqrt{71}.

$$x=\frac{2\sqrt{71}+12}{10}$$

Whakawehe 12+2\sqrt{71} ki te 10.

$$x=\frac{\sqrt{71}+6}{5}$$

Nā, me whakaoti te whārite x=\frac{12±2\sqrt{71}}{10} ina he tango te ±. Tango 2\sqrt{71} mai i 12.

$$x=\frac{12-2\sqrt{71}}{10}$$

Whakawehe 12-2\sqrt{71} ki te 10.

$$x=\frac{6-\sqrt{71}}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{71}+6}{5} x=\frac{6-\sqrt{71}}{5}$$

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.

$$5x^{2}-12x-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$5x^{2}-12x-7-\left(-7\right)=-\left(-7\right)$$

Mā te tango i te -7 i a ia ake anō ka toe ko te 0.

$$5x^{2}-12x=-\left(-7\right)$$

Tango -7 mai i 0.

$$5x^{2}-12x=7$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-12x}{5}=\frac{7}{5}$$

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

$$x^{2}-\frac{12}{5}x=\frac{7}{5}$$

Whakawehea te -\frac{12}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{6}{5}. Nā, tāpiria te pūrua o te -\frac{6}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{12}{5}x+\left(-\frac{6}{5}\right)^{2}=\frac{7}{5}+\left(-\frac{6}{5}\right)^{2}$$

Pūruatia -\frac{6}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{12}{5}x+\frac{36}{25}=\frac{7}{5}+\frac{36}{25}$$

Tāpiri \frac{7}{5} ki te \frac{36}{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.

$$x^{2}-\frac{12}{5}x+\frac{36}{25}=\frac{71}{25}$$

Tauwehea x^{2}-\frac{12}{5}x+\frac{36}{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}.

$$\left(x-\frac{6}{5}\right)^{2}=\frac{71}{25}$$

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

$$\sqrt{\left(x-\frac{6}{5}\right)^{2}}=\sqrt{\frac{71}{25}}$$

Whakarūnātia.

$$x-\frac{6}{5}=\frac{\sqrt{71}}{5} x-\frac{6}{5}=-\frac{\sqrt{71}}{5}$$

Me tāpiri \frac{6}{5} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{71}+6}{5} x=\frac{6-\sqrt{71}}{5}$$