Whakaoti i te 21x^2-6x=13

$$21 x ^ { 2 } - 6 x = 13$$

Whakaoti mō x

$$x=\frac{\sqrt{282}}{21}+\frac{1}{7}\approx 0.942516934$$

$$x=-\frac{\sqrt{282}}{21}+\frac{1}{7}\approx -0.656802649$$

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.

$$21x^{2}-6x=13$$

Me tango 13 mai i ngā taha e rua o te whārite.

$$21x^{2}-6x-13=13-13$$

Mā te tango i te 13 i a ia ake anō ka toe ko te 0.

$$21x^{2}-6x-13=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 21 mō a, -6 mō b, me -13 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 21\left(-13\right)}}{2\times 21}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 21\left(-13\right)}}{2\times 21}$$

Whakareatia -4 ki te 21.

$$x=\frac{-\left(-6\right)±\sqrt{36-84\left(-13\right)}}{2\times 21}$$

Whakareatia -84 ki te -13.

$$x=\frac{-\left(-6\right)±\sqrt{36+1092}}{2\times 21}$$

Tāpiri 36 ki te 1092.

$$x=\frac{-\left(-6\right)±\sqrt{1128}}{2\times 21}$$

Tuhia te pūtakerua o te 1128.

$$x=\frac{-\left(-6\right)±2\sqrt{282}}{2\times 21}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±2\sqrt{282}}{2\times 21}$$

Whakareatia 2 ki te 21.

$$x=\frac{6±2\sqrt{282}}{42}$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{282}}{42} ina he tāpiri te ±. Tāpiri 6 ki te 2\sqrt{282}.

$$x=\frac{2\sqrt{282}+6}{42}$$

Whakawehe 6+2\sqrt{282} ki te 42.

$$x=\frac{\sqrt{282}}{21}+\frac{1}{7}$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{282}}{42} ina he tango te ±. Tango 2\sqrt{282} mai i 6.

$$x=\frac{6-2\sqrt{282}}{42}$$

Whakawehe 6-2\sqrt{282} ki te 42.

$$x=-\frac{\sqrt{282}}{21}+\frac{1}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{282}}{21}+\frac{1}{7} x=-\frac{\sqrt{282}}{21}+\frac{1}{7}$$

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.

$$21x^{2}-6x=13$$

Whakawehea ngā taha e rua ki te 21.

$$\frac{21x^{2}-6x}{21}=\frac{13}{21}$$

Mā te whakawehe ki te 21 ka wetekia te whakareanga ki te 21.

$$x^{2}+\left(-\frac{6}{21}\right)x=\frac{13}{21}$$

Whakahekea te hautanga \frac{-6}{21} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{2}{7}x=\frac{13}{21}$$

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.

$$x^{2}-\frac{2}{7}x+\left(-\frac{1}{7}\right)^{2}=\frac{13}{21}+\left(-\frac{1}{7}\right)^{2}$$

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

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{13}{21}+\frac{1}{49}$$

Tāpiri \frac{13}{21} 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.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{94}{147}$$

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}.

$$\left(x-\frac{1}{7}\right)^{2}=\frac{94}{147}$$

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

$$\sqrt{\left(x-\frac{1}{7}\right)^{2}}=\sqrt{\frac{94}{147}}$$

Whakarūnātia.

$$x-\frac{1}{7}=\frac{\sqrt{282}}{21} x-\frac{1}{7}=-\frac{\sqrt{282}}{21}$$

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

$$x=\frac{\sqrt{282}}{21}+\frac{1}{7} x=-\frac{\sqrt{282}}{21}+\frac{1}{7}$$