Whakaoti i te 2x^2-14x-54=0

$$2 { x }^{ 2 } -14x-54=0$$

Whakaoti mō x

$$x = \frac{\sqrt{157} + 7}{2} \approx 9.764982043$$

$$x=\frac{7-\sqrt{157}}{2}\approx -2.764982043$$

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.

$$2x^{2}-14x-54=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -14 mō b, me -54 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\left(-54\right)}}{2\times 2}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 2\left(-54\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-14\right)±\sqrt{196-8\left(-54\right)}}{2\times 2}$$

Whakareatia -8 ki te -54.

$$x=\frac{-\left(-14\right)±\sqrt{196+432}}{2\times 2}$$

Tāpiri 196 ki te 432.

$$x=\frac{-\left(-14\right)±\sqrt{628}}{2\times 2}$$

Tuhia te pūtakerua o te 628.

$$x=\frac{-\left(-14\right)±2\sqrt{157}}{2\times 2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±2\sqrt{157}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{14±2\sqrt{157}}{4}$$

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{157}}{4} ina he tāpiri te ±. Tāpiri 14 ki te 2\sqrt{157}.

$$x=\frac{2\sqrt{157}+14}{4}$$

Whakawehe 14+2\sqrt{157} ki te 4.

$$x=\frac{\sqrt{157}+7}{2}$$

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{157}}{4} ina he tango te ±. Tango 2\sqrt{157} mai i 14.

$$x=\frac{14-2\sqrt{157}}{4}$$

Whakawehe 14-2\sqrt{157} ki te 4.

$$x=\frac{7-\sqrt{157}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{157}+7}{2} x=\frac{7-\sqrt{157}}{2}$$

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.

$$2x^{2}-14x-54=0$$

Me tāpiri 54 ki ngā taha e rua o te whārite.

$$2x^{2}-14x-54-\left(-54\right)=-\left(-54\right)$$

Mā te tango i te -54 i a ia ake anō ka toe ko te 0.

$$2x^{2}-14x=-\left(-54\right)$$

Tango -54 mai i 0.

$$2x^{2}-14x=54$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-14x}{2}=\frac{54}{2}$$

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

$$x^{2}+\left(-\frac{14}{2}\right)x=\frac{54}{2}$$

Whakawehe -14 ki te 2.

$$x^{2}-7x=\frac{54}{2}$$

Whakawehe 54 ki te 2.

$$x^{2}-7x=27$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=27+\left(-\frac{7}{2}\right)^{2}$$

Pūruatia -\frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-7x+\frac{49}{4}=27+\frac{49}{4}$$

Tāpiri 27 ki te \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{157}{4}$$

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

$$\left(x-\frac{7}{2}\right)^{2}=\frac{157}{4}$$

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

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{157}{4}}$$

Whakarūnātia.

$$x-\frac{7}{2}=\frac{\sqrt{157}}{2} x-\frac{7}{2}=-\frac{\sqrt{157}}{2}$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{157}+7}{2} x=\frac{7-\sqrt{157}}{2}$$