Whakaoti i te 3x^2-40x+96=0

$$3 x ^ { 2 } - 40 x + 96 = 0$$

Whakaoti mō x

$$x = \frac{4 \sqrt{7} + 20}{3} \approx 10.194335081$$

$$x = \frac{20 - 4 \sqrt{7}}{3} \approx 3.138998252$$

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.

$$3x^{2}-40x+96=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -40 mō b, me 96 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 3\times 96}}{2\times 3}$$

Pūrua -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 3\times 96}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-40\right)±\sqrt{1600-12\times 96}}{2\times 3}$$

Whakareatia -12 ki te 96.

$$x=\frac{-\left(-40\right)±\sqrt{1600-1152}}{2\times 3}$$

Tāpiri 1600 ki te -1152.

$$x=\frac{-\left(-40\right)±\sqrt{448}}{2\times 3}$$

Tuhia te pūtakerua o te 448.

$$x=\frac{-\left(-40\right)±8\sqrt{7}}{2\times 3}$$

Ko te tauaro o -40 ko 40.

$$x=\frac{40±8\sqrt{7}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{40±8\sqrt{7}}{6}$$

Nā, me whakaoti te whārite x=\frac{40±8\sqrt{7}}{6} ina he tāpiri te ±. Tāpiri 40 ki te 8\sqrt{7}.

$$x=\frac{8\sqrt{7}+40}{6}$$

Whakawehe 40+8\sqrt{7} ki te 6.

$$x=\frac{4\sqrt{7}+20}{3}$$

Nā, me whakaoti te whārite x=\frac{40±8\sqrt{7}}{6} ina he tango te ±. Tango 8\sqrt{7} mai i 40.

$$x=\frac{40-8\sqrt{7}}{6}$$

Whakawehe 40-8\sqrt{7} ki te 6.

$$x=\frac{20-4\sqrt{7}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{4\sqrt{7}+20}{3} x=\frac{20-4\sqrt{7}}{3}$$

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.

$$3x^{2}-40x+96=0$$

Me tango 96 mai i ngā taha e rua o te whārite.

$$3x^{2}-40x+96-96=-96$$

Mā te tango i te 96 i a ia ake anō ka toe ko te 0.

$$3x^{2}-40x=-96$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-40x}{3}=-\frac{96}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{40}{3}x=-\frac{96}{3}$$

Whakawehe -96 ki te 3.

$$x^{2}-\frac{40}{3}x=-32$$

Whakawehea te -\frac{40}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{20}{3}. Nā, tāpiria te pūrua o te -\frac{20}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{40}{3}x+\left(-\frac{20}{3}\right)^{2}=-32+\left(-\frac{20}{3}\right)^{2}$$

Pūruatia -\frac{20}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{40}{3}x+\frac{400}{9}=-32+\frac{400}{9}$$

Tāpiri -32 ki te \frac{400}{9}.

$$x^{2}-\frac{40}{3}x+\frac{400}{9}=\frac{112}{9}$$

Tauwehea x^{2}-\frac{40}{3}x+\frac{400}{9}. 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}.

$$\left(x-\frac{20}{3}\right)^{2}=\frac{112}{9}$$

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

$$\sqrt{\left(x-\frac{20}{3}\right)^{2}}=\sqrt{\frac{112}{9}}$$

Whakarūnātia.

$$x-\frac{20}{3}=\frac{4\sqrt{7}}{3} x-\frac{20}{3}=-\frac{4\sqrt{7}}{3}$$

Me tāpiri \frac{20}{3} ki ngā taha e rua o te whārite.

$$x=\frac{4\sqrt{7}+20}{3} x=\frac{20-4\sqrt{7}}{3}$$