Whakaoti i te 20x^2-28x-1=0

$$20 x ^ { 2 } - 28 x - 1 = 0$$

Whakaoti mō x

$$x = \frac{3 \sqrt{6} + 7}{10} \approx 1.434846923$$

$$x=\frac{7-3\sqrt{6}}{10}\approx -0.034846923$$

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.

$$20x^{2}-28x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 20 mō a, -28 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 20\left(-1\right)}}{2\times 20}$$

Pūrua -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 20\left(-1\right)}}{2\times 20}$$

Whakareatia -4 ki te 20.

$$x=\frac{-\left(-28\right)±\sqrt{784-80\left(-1\right)}}{2\times 20}$$

Whakareatia -80 ki te -1.

$$x=\frac{-\left(-28\right)±\sqrt{784+80}}{2\times 20}$$

Tāpiri 784 ki te 80.

$$x=\frac{-\left(-28\right)±\sqrt{864}}{2\times 20}$$

Tuhia te pūtakerua o te 864.

$$x=\frac{-\left(-28\right)±12\sqrt{6}}{2\times 20}$$

Ko te tauaro o -28 ko 28.

$$x=\frac{28±12\sqrt{6}}{2\times 20}$$

Whakareatia 2 ki te 20.

$$x=\frac{28±12\sqrt{6}}{40}$$

Nā, me whakaoti te whārite x=\frac{28±12\sqrt{6}}{40} ina he tāpiri te ±. Tāpiri 28 ki te 12\sqrt{6}.

$$x=\frac{12\sqrt{6}+28}{40}$$

Whakawehe 28+12\sqrt{6} ki te 40.

$$x=\frac{3\sqrt{6}+7}{10}$$

Nā, me whakaoti te whārite x=\frac{28±12\sqrt{6}}{40} ina he tango te ±. Tango 12\sqrt{6} mai i 28.

$$x=\frac{28-12\sqrt{6}}{40}$$

Whakawehe 28-12\sqrt{6} ki te 40.

$$x=\frac{7-3\sqrt{6}}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{6}+7}{10} x=\frac{7-3\sqrt{6}}{10}$$

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.

$$20x^{2}-28x-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$20x^{2}-28x-1-\left(-1\right)=-\left(-1\right)$$

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

$$20x^{2}-28x=-\left(-1\right)$$

Tango -1 mai i 0.

$$20x^{2}-28x=1$$

Whakawehea ngā taha e rua ki te 20.

$$\frac{20x^{2}-28x}{20}=\frac{1}{20}$$

Mā te whakawehe ki te 20 ka wetekia te whakareanga ki te 20.

$$x^{2}+\left(-\frac{28}{20}\right)x=\frac{1}{20}$$

Whakahekea te hautanga \frac{-28}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{7}{5}x=\frac{1}{20}$$

Whakawehea te -\frac{7}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{10}. Nā, tāpiria te pūrua o te -\frac{7}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{5}x+\left(-\frac{7}{10}\right)^{2}=\frac{1}{20}+\left(-\frac{7}{10}\right)^{2}$$

Pūruatia -\frac{7}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{1}{20}+\frac{49}{100}$$

Tāpiri \frac{1}{20} ki te \frac{49}{100} 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.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{27}{50}$$

Tauwehea x^{2}-\frac{7}{5}x+\frac{49}{100}. 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}.

$$\left(x-\frac{7}{10}\right)^{2}=\frac{27}{50}$$

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

$$\sqrt{\left(x-\frac{7}{10}\right)^{2}}=\sqrt{\frac{27}{50}}$$

Whakarūnātia.

$$x-\frac{7}{10}=\frac{3\sqrt{6}}{10} x-\frac{7}{10}=-\frac{3\sqrt{6}}{10}$$

Me tāpiri \frac{7}{10} ki ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{6}+7}{10} x=\frac{7-3\sqrt{6}}{10}$$