Whakaoti i te 5x^2+18x+1=0

$$5 x ^ { 2 } + 18 x + 1 = 0$$

Whakaoti mō x

$$x=\frac{2\sqrt{19}-9}{5}\approx -0.056440423$$

$$x=\frac{-2\sqrt{19}-9}{5}\approx -3.543559577$$

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.

$$5x^{2}+18x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 18 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 5}}{2\times 5}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 5}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-18±\sqrt{324-20}}{2\times 5}$$

Tāpiri 324 ki te -20.

$$x=\frac{-18±\sqrt{304}}{2\times 5}$$

Tuhia te pūtakerua o te 304.

$$x=\frac{-18±4\sqrt{19}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-18±4\sqrt{19}}{10}$$

Nā, me whakaoti te whārite x=\frac{-18±4\sqrt{19}}{10} ina he tāpiri te ±. Tāpiri -18 ki te 4\sqrt{19}.

$$x=\frac{4\sqrt{19}-18}{10}$$

Whakawehe -18+4\sqrt{19} ki te 10.

$$x=\frac{2\sqrt{19}-9}{5}$$

Nā, me whakaoti te whārite x=\frac{-18±4\sqrt{19}}{10} ina he tango te ±. Tango 4\sqrt{19} mai i -18.

$$x=\frac{-4\sqrt{19}-18}{10}$$

Whakawehe -18-4\sqrt{19} ki te 10.

$$x=\frac{-2\sqrt{19}-9}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{19}-9}{5} x=\frac{-2\sqrt{19}-9}{5}$$

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.

$$5x^{2}+18x+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$5x^{2}+18x+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$5x^{2}+18x=-1$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+18x}{5}=-\frac{1}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\frac{18}{5}x=-\frac{1}{5}$$

Whakawehea te \frac{18}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{5}. Nā, tāpiria te pūrua o te \frac{9}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{18}{5}x+\left(\frac{9}{5}\right)^{2}=-\frac{1}{5}+\left(\frac{9}{5}\right)^{2}$$

Pūruatia \frac{9}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{18}{5}x+\frac{81}{25}=-\frac{1}{5}+\frac{81}{25}$$

Tāpiri -\frac{1}{5} ki te \frac{81}{25} 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.

$$x^{2}+\frac{18}{5}x+\frac{81}{25}=\frac{76}{25}$$

Tauwehea x^{2}+\frac{18}{5}x+\frac{81}{25}. 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}.

$$\left(x+\frac{9}{5}\right)^{2}=\frac{76}{25}$$

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

$$\sqrt{\left(x+\frac{9}{5}\right)^{2}}=\sqrt{\frac{76}{25}}$$

Whakarūnātia.

$$x+\frac{9}{5}=\frac{2\sqrt{19}}{5} x+\frac{9}{5}=-\frac{2\sqrt{19}}{5}$$

Me tango \frac{9}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{19}-9}{5} x=\frac{-2\sqrt{19}-9}{5}$$