Whakaoti i te 2x^2-7x+4=0

$$2 x ^ { 2 } - 7 x + 4 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{17} + 7}{4} \approx 2.780776406$$

$$x=\frac{7-\sqrt{17}}{4}\approx 0.719223594$$

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.

$$2x^{2}-7x+4=0$$

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

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

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 2\times 4}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-8\times 4}}{2\times 2}$$

Whakareatia -8 ki te 4.

$$x=\frac{-\left(-7\right)±\sqrt{49-32}}{2\times 2}$$

Tāpiri 49 ki te -32.

$$x=\frac{-\left(-7\right)±\sqrt{17}}{2\times 2}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{17}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{7±\sqrt{17}}{4}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{17}}{4} ina he tāpiri te ±. Tāpiri 7 ki te \sqrt{17}.

$$x=\frac{\sqrt{17}+7}{4}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{17}}{4} ina he tango te ±. Tango \sqrt{17} mai i 7.

$$x=\frac{7-\sqrt{17}}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{17}+7}{4} x=\frac{7-\sqrt{17}}{4}$$

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.

$$2x^{2}-7x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$2x^{2}-7x+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$2x^{2}-7x=-4$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakawehe -4 ki te 2.

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

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-2+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-2+\frac{49}{16}$$

Tāpiri -2 ki te \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{17}{16}$$

Tauwehea x^{2}-\frac{7}{2}x+\frac{49}{16}. 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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{17}{16}$$

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

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Whakarūnātia.

$$x-\frac{7}{4}=\frac{\sqrt{17}}{4} x-\frac{7}{4}=-\frac{\sqrt{17}}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{17}+7}{4} x=\frac{7-\sqrt{17}}{4}$$