Whakaoti i te 7x^2-3x-5=0

$$7 { x }^{ 2 } -3x-5=0$$

Whakaoti mō x

$$x = \frac{\sqrt{149} + 3}{14} \approx 1.086182544$$

$$x=\frac{3-\sqrt{149}}{14}\approx -0.657611115$$

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.

$$7x^{2}-3x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -3 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 7\left(-5\right)}}{2\times 7}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 7\left(-5\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-3\right)±\sqrt{9-28\left(-5\right)}}{2\times 7}$$

Whakareatia -28 ki te -5.

$$x=\frac{-\left(-3\right)±\sqrt{9+140}}{2\times 7}$$

Tāpiri 9 ki te 140.

$$x=\frac{-\left(-3\right)±\sqrt{149}}{2\times 7}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±\sqrt{149}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{3±\sqrt{149}}{14}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{149}}{14} ina he tāpiri te ±. Tāpiri 3 ki te \sqrt{149}.

$$x=\frac{\sqrt{149}+3}{14}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{149}}{14} ina he tango te ±. Tango \sqrt{149} mai i 3.

$$x=\frac{3-\sqrt{149}}{14}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{149}+3}{14} x=\frac{3-\sqrt{149}}{14}$$

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.

$$7x^{2}-3x-5=0$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$7x^{2}-3x-5-\left(-5\right)=-\left(-5\right)$$

Mā te tango i te -5 i a ia ake anō ka toe ko te 0.

$$7x^{2}-3x=-\left(-5\right)$$

Tango -5 mai i 0.

$$7x^{2}-3x=5$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-3x}{7}=\frac{5}{7}$$

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

$$x^{2}-\frac{3}{7}x=\frac{5}{7}$$

Whakawehea te -\frac{3}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{14}. Nā, tāpiria te pūrua o te -\frac{3}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{7}x+\left(-\frac{3}{14}\right)^{2}=\frac{5}{7}+\left(-\frac{3}{14}\right)^{2}$$

Pūruatia -\frac{3}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{5}{7}+\frac{9}{196}$$

Tāpiri \frac{5}{7} ki te \frac{9}{196} 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.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{149}{196}$$

Tauwehea te x^{2}-\frac{3}{7}x+\frac{9}{196}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{14}\right)^{2}=\frac{149}{196}$$

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

$$\sqrt{\left(x-\frac{3}{14}\right)^{2}}=\sqrt{\frac{149}{196}}$$

Whakarūnātia.

$$x-\frac{3}{14}=\frac{\sqrt{149}}{14} x-\frac{3}{14}=-\frac{\sqrt{149}}{14}$$

Me tāpiri \frac{3}{14} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{149}+3}{14} x=\frac{3-\sqrt{149}}{14}$$