Whakaoti i te 2x^2-55x+3=0

$$2 { x }^{ 2 } -55x+3=0$$

Whakaoti mō x

$$x = \frac{\sqrt{3001} + 55}{4} \approx 27.445345925$$

$$x=\frac{55-\sqrt{3001}}{4}\approx 0.054654075$$

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.

$$2x^{2}-55x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -55 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\times 2\times 3}}{2\times 2}$$

Pūrua -55.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\times 2\times 3}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-55\right)±\sqrt{3025-8\times 3}}{2\times 2}$$

Whakareatia -8 ki te 3.

$$x=\frac{-\left(-55\right)±\sqrt{3025-24}}{2\times 2}$$

Tāpiri 3025 ki te -24.

$$x=\frac{-\left(-55\right)±\sqrt{3001}}{2\times 2}$$

Ko te tauaro o -55 ko 55.

$$x=\frac{55±\sqrt{3001}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{55±\sqrt{3001}}{4}$$

Nā, me whakaoti te whārite x=\frac{55±\sqrt{3001}}{4} ina he tāpiri te ±. Tāpiri 55 ki te \sqrt{3001}.

$$x=\frac{\sqrt{3001}+55}{4}$$

Nā, me whakaoti te whārite x=\frac{55±\sqrt{3001}}{4} ina he tango te ±. Tango \sqrt{3001} mai i 55.

$$x=\frac{55-\sqrt{3001}}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{3001}+55}{4} x=\frac{55-\sqrt{3001}}{4}$$

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.

$$2x^{2}-55x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$2x^{2}-55x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$2x^{2}-55x=-3$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-55x}{2}=-\frac{3}{2}$$

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

$$x^{2}-\frac{55}{2}x=-\frac{3}{2}$$

Whakawehea te -\frac{55}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{55}{4}. Nā, tāpiria te pūrua o te -\frac{55}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{55}{2}x+\left(-\frac{55}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{55}{4}\right)^{2}$$

Pūruatia -\frac{55}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{55}{2}x+\frac{3025}{16}=-\frac{3}{2}+\frac{3025}{16}$$

Tāpiri -\frac{3}{2} ki te \frac{3025}{16} 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.

$$x^{2}-\frac{55}{2}x+\frac{3025}{16}=\frac{3001}{16}$$

Tauwehea x^{2}-\frac{55}{2}x+\frac{3025}{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}.

$$\left(x-\frac{55}{4}\right)^{2}=\frac{3001}{16}$$

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

$$\sqrt{\left(x-\frac{55}{4}\right)^{2}}=\sqrt{\frac{3001}{16}}$$

Whakarūnātia.

$$x-\frac{55}{4}=\frac{\sqrt{3001}}{4} x-\frac{55}{4}=-\frac{\sqrt{3001}}{4}$$

Me tāpiri \frac{55}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{3001}+55}{4} x=\frac{55-\sqrt{3001}}{4}$$