Whakaoti i te 2x^2-10x+7=0

$$2 x ^ { 2 } - 10 x + 7 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{11} + 5}{2} \approx 4.158312395$$

$$x=\frac{5-\sqrt{11}}{2}\approx 0.841687605$$

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.

$$2x^{2}-10x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -10 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 2\times 7}}{2\times 2}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 2\times 7}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-10\right)±\sqrt{100-8\times 7}}{2\times 2}$$

Whakareatia -8 ki te 7.

$$x=\frac{-\left(-10\right)±\sqrt{100-56}}{2\times 2}$$

Tāpiri 100 ki te -56.

$$x=\frac{-\left(-10\right)±\sqrt{44}}{2\times 2}$$

Tuhia te pūtakerua o te 44.

$$x=\frac{-\left(-10\right)±2\sqrt{11}}{2\times 2}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±2\sqrt{11}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{10±2\sqrt{11}}{4}$$

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{11}}{4} ina he tāpiri te ±. Tāpiri 10 ki te 2\sqrt{11}.

$$x=\frac{2\sqrt{11}+10}{4}$$

Whakawehe 10+2\sqrt{11} ki te 4.

$$x=\frac{\sqrt{11}+5}{2}$$

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{11}}{4} ina he tango te ±. Tango 2\sqrt{11} mai i 10.

$$x=\frac{10-2\sqrt{11}}{4}$$

Whakawehe 10-2\sqrt{11} ki te 4.

$$x=\frac{5-\sqrt{11}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{11}+5}{2} x=\frac{5-\sqrt{11}}{2}$$

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.

$$2x^{2}-10x+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$2x^{2}-10x+7-7=-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$2x^{2}-10x=-7$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-10x}{2}=-\frac{7}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

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

Whakawehe -10 ki te 2.

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

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

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

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=-\frac{7}{2}+\frac{25}{4}$$

Tāpiri -\frac{7}{2} ki te \frac{25}{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.

$$x^{2}-5x+\frac{25}{4}=\frac{11}{4}$$

Tauwehea x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{11}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{11}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{\sqrt{11}}{2} x-\frac{5}{2}=-\frac{\sqrt{11}}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{11}+5}{2} x=\frac{5-\sqrt{11}}{2}$$