Whakaoti i te 2+3x-4x^2=0

$$2 + 3 x - 4 x ^ { 2 } = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{41} + 3}{8} \approx 1.17539053$$

$$x=\frac{3-\sqrt{41}}{8}\approx -0.42539053$$

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.

$$-4x^{2}+3x+2=0$$

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

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

Pūrua 3.

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

Whakareatia -4 ki te -4.

$$x=\frac{-3±\sqrt{9+16\times 2}}{2\left(-4\right)}$$

Whakareatia 16 ki te 2.

$$x=\frac{-3±\sqrt{9+32}}{2\left(-4\right)}$$

Tāpiri 9 ki te 32.

$$x=\frac{-3±\sqrt{41}}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$x=\frac{-3±\sqrt{41}}{-8}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{41}}{-8} ina he tāpiri te ±. Tāpiri -3 ki te \sqrt{41}.

$$x=\frac{\sqrt{41}-3}{-8}$$

Whakawehe -3+\sqrt{41} ki te -8.

$$x=\frac{3-\sqrt{41}}{8}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{41}}{-8} ina he tango te ±. Tango \sqrt{41} mai i -3.

$$x=\frac{-\sqrt{41}-3}{-8}$$

Whakawehe -3-\sqrt{41} ki te -8.

$$x=\frac{\sqrt{41}+3}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{3-\sqrt{41}}{8} x=\frac{\sqrt{41}+3}{8}$$

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.

$$-4x^{2}+3x+2=0$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$-4x^{2}+3x+2-2=-2$$

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

$$-4x^{2}+3x=-2$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4x^{2}+3x}{-4}=-\frac{2}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$x^{2}+\frac{3}{-4}x=-\frac{2}{-4}$$

Whakawehe 3 ki te -4.

$$x^{2}-\frac{3}{4}x=-\frac{2}{-4}$$

Whakahekea te hautanga \frac{-2}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{3}{4}x=\frac{1}{2}$$

Whakawehea te -\frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{8}. Nā, tāpiria te pūrua o te -\frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=\frac{1}{2}+\left(-\frac{3}{8}\right)^{2}$$

Pūruatia -\frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{1}{2}+\frac{9}{64}$$

Tāpiri \frac{1}{2} ki te \frac{9}{64} 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.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{41}{64}$$

Tauwehea x^{2}-\frac{3}{4}x+\frac{9}{64}. 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}.

$$\left(x-\frac{3}{8}\right)^{2}=\frac{41}{64}$$

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

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{\frac{41}{64}}$$

Whakarūnātia.

$$x-\frac{3}{8}=\frac{\sqrt{41}}{8} x-\frac{3}{8}=-\frac{\sqrt{41}}{8}$$

Me tāpiri \frac{3}{8} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{41}+3}{8} x=\frac{3-\sqrt{41}}{8}$$