Whakaoti i te 2x^2-34x=-22

$$2 x ^ { 2 } - 34 x = - 22$$

Whakaoti mō x

$$x = \frac{7 \sqrt{5} + 17}{2} \approx 16.326237921$$

$$x=\frac{17-7\sqrt{5}}{2}\approx 0.673762079$$

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.

$$2x^{2}-34x=-22$$

Me tāpiri 22 ki ngā taha e rua o te whārite.

$$2x^{2}-34x-\left(-22\right)=-22-\left(-22\right)$$

Mā te tango i te -22 i a ia ake anō ka toe ko te 0.

$$2x^{2}-34x-\left(-22\right)=0$$

Tango -22 mai i 0.

$$2x^{2}-34x+22=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -34 mō b, me 22 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 2\times 22}}{2\times 2}$$

Pūrua -34.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 2\times 22}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-34\right)±\sqrt{1156-8\times 22}}{2\times 2}$$

Whakareatia -8 ki te 22.

$$x=\frac{-\left(-34\right)±\sqrt{1156-176}}{2\times 2}$$

Tāpiri 1156 ki te -176.

$$x=\frac{-\left(-34\right)±\sqrt{980}}{2\times 2}$$

Tuhia te pūtakerua o te 980.

$$x=\frac{-\left(-34\right)±14\sqrt{5}}{2\times 2}$$

Ko te tauaro o -34 ko 34.

$$x=\frac{34±14\sqrt{5}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{34±14\sqrt{5}}{4}$$

Nā, me whakaoti te whārite x=\frac{34±14\sqrt{5}}{4} ina he tāpiri te ±. Tāpiri 34 ki te 14\sqrt{5}.

$$x=\frac{14\sqrt{5}+34}{4}$$

Whakawehe 34+14\sqrt{5} ki te 4.

$$x=\frac{7\sqrt{5}+17}{2}$$

Nā, me whakaoti te whārite x=\frac{34±14\sqrt{5}}{4} ina he tango te ±. Tango 14\sqrt{5} mai i 34.

$$x=\frac{34-14\sqrt{5}}{4}$$

Whakawehe 34-14\sqrt{5} ki te 4.

$$x=\frac{17-7\sqrt{5}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{7\sqrt{5}+17}{2} x=\frac{17-7\sqrt{5}}{2}$$

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.

$$2x^{2}-34x=-22$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-34x}{2}=-\frac{22}{2}$$

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

$$x^{2}+\left(-\frac{34}{2}\right)x=-\frac{22}{2}$$

Whakawehe -34 ki te 2.

$$x^{2}-17x=-\frac{22}{2}$$

Whakawehe -22 ki te 2.

$$x^{2}-17x=-11$$

Whakawehea te -17, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{17}{2}. Nā, tāpiria te pūrua o te -\frac{17}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-11+\left(-\frac{17}{2}\right)^{2}$$

Pūruatia -\frac{17}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-17x+\frac{289}{4}=-11+\frac{289}{4}$$

Tāpiri -11 ki te \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{245}{4}$$

Tauwehea x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{245}{4}$$

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

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{245}{4}}$$

Whakarūnātia.

$$x-\frac{17}{2}=\frac{7\sqrt{5}}{2} x-\frac{17}{2}=-\frac{7\sqrt{5}}{2}$$

Me tāpiri \frac{17}{2} ki ngā taha e rua o te whārite.

$$x=\frac{7\sqrt{5}+17}{2} x=\frac{17-7\sqrt{5}}{2}$$