Whakaoti i te x^2-x=120/7

$$x ^ { 2 } - x = \frac { 120 } { 7 }$$

Whakaoti mō x

$$x=\frac{\sqrt{3409}}{14}+\frac{1}{2}\approx 4.670474451$$

$$x=-\frac{\sqrt{3409}}{14}+\frac{1}{2}\approx -3.670474451$$

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.

$$x^{2}-x=\frac{120}{7}$$

Me tango \frac{120}{7} mai i ngā taha e rua o te whārite.

$$x^{2}-x-\frac{120}{7}=\frac{120}{7}-\frac{120}{7}$$

Mā te tango i te \frac{120}{7} i a ia ake anō ka toe ko te 0.

$$x^{2}-x-\frac{120}{7}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -\frac{120}{7} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-\frac{120}{7}\right)}}{2}$$

Whakareatia -4 ki te -\frac{120}{7}.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{480}{7}}}{2}$$

Tāpiri 1 ki te \frac{480}{7}.

$$x=\frac{-\left(-1\right)±\sqrt{\frac{487}{7}}}{2}$$

Tuhia te pūtakerua o te \frac{487}{7}.

$$x=\frac{-\left(-1\right)±\frac{\sqrt{3409}}{7}}{2}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\frac{\sqrt{3409}}{7}}{2}$$

Nā, me whakaoti te whārite x=\frac{1±\frac{\sqrt{3409}}{7}}{2} ina he tāpiri te ±. Tāpiri 1 ki te \frac{\sqrt{3409}}{7}.

$$x=\frac{\frac{\sqrt{3409}}{7}+1}{2}$$

Whakawehe 1+\frac{\sqrt{3409}}{7} ki te 2.

$$x=\frac{\sqrt{3409}}{14}+\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{1±\frac{\sqrt{3409}}{7}}{2} ina he tango te ±. Tango \frac{\sqrt{3409}}{7} mai i 1.

$$x=\frac{-\frac{\sqrt{3409}}{7}+1}{2}$$

Whakawehe 1-\frac{\sqrt{3409}}{7} ki te 2.

$$x=-\frac{\sqrt{3409}}{14}+\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{3409}}{14}+\frac{1}{2} x=-\frac{\sqrt{3409}}{14}+\frac{1}{2}$$

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.

$$x^{2}-x=\frac{120}{7}$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{120}{7}+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=\frac{120}{7}+\frac{1}{4}$$

Tāpiri \frac{120}{7} ki te \frac{1}{4} 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.

$$x^{2}-x+\frac{1}{4}=\frac{487}{28}$$

Tauwehea x^{2}-x+\frac{1}{4}. 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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{487}{28}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{487}{28}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{3409}}{14} x-\frac{1}{2}=-\frac{\sqrt{3409}}{14}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{3409}}{14}+\frac{1}{2} x=-\frac{\sqrt{3409}}{14}+\frac{1}{2}$$