Whakaoti i te 7x^2-2x-3=0

$$7 x ^ { 2 } - 2 x - 3 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{22}+1}{7}\approx 0.812916537$$

$$x=\frac{1-\sqrt{22}}{7}\approx -0.527202251$$

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.

$$7x^{2}-2x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -2 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 7\left(-3\right)}}{2\times 7}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 7\left(-3\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-2\right)±\sqrt{4-28\left(-3\right)}}{2\times 7}$$

Whakareatia -28 ki te -3.

$$x=\frac{-\left(-2\right)±\sqrt{4+84}}{2\times 7}$$

Tāpiri 4 ki te 84.

$$x=\frac{-\left(-2\right)±\sqrt{88}}{2\times 7}$$

Tuhia te pūtakerua o te 88.

$$x=\frac{-\left(-2\right)±2\sqrt{22}}{2\times 7}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±2\sqrt{22}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{2±2\sqrt{22}}{14}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{22}}{14} ina he tāpiri te ±. Tāpiri 2 ki te 2\sqrt{22}.

$$x=\frac{2\sqrt{22}+2}{14}$$

Whakawehe 2+2\sqrt{22} ki te 14.

$$x=\frac{\sqrt{22}+1}{7}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{22}}{14} ina he tango te ±. Tango 2\sqrt{22} mai i 2.

$$x=\frac{2-2\sqrt{22}}{14}$$

Whakawehe 2-2\sqrt{22} ki te 14.

$$x=\frac{1-\sqrt{22}}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{22}+1}{7} x=\frac{1-\sqrt{22}}{7}$$

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.

$$7x^{2}-2x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$7x^{2}-2x-3-\left(-3\right)=-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$7x^{2}-2x=-\left(-3\right)$$

Tango -3 mai i 0.

$$7x^{2}-2x=3$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-2x}{7}=\frac{3}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}-\frac{2}{7}x=\frac{3}{7}$$

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.

$$x^{2}-\frac{2}{7}x+\left(-\frac{1}{7}\right)^{2}=\frac{3}{7}+\left(-\frac{1}{7}\right)^{2}$$

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

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{3}{7}+\frac{1}{49}$$

Tāpiri \frac{3}{7} 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.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{22}{49}$$

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}.

$$\left(x-\frac{1}{7}\right)^{2}=\frac{22}{49}$$

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

$$\sqrt{\left(x-\frac{1}{7}\right)^{2}}=\sqrt{\frac{22}{49}}$$

Whakarūnātia.

$$x-\frac{1}{7}=\frac{\sqrt{22}}{7} x-\frac{1}{7}=-\frac{\sqrt{22}}{7}$$

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

$$x=\frac{\sqrt{22}+1}{7} x=\frac{1-\sqrt{22}}{7}$$