Whakaoti i te 9x^2+5x+3=20

$$9 x ^ { 2 } + 5 x + 3 = 20$$

Whakaoti mō x

$$x = \frac{7 \sqrt{13} - 5}{18} \approx 1.124381052$$

$$x=\frac{-7\sqrt{13}-5}{18}\approx -1.679936607$$

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.

$$9x^{2}+5x+3=20$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$9x^{2}+5x+3-20=20-20$$

Mā te tango i te 20 i a ia ake anō ka toe ko te 0.

$$9x^{2}+5x+3-20=0$$

Tango 20 mai i 3.

$$9x^{2}+5x-17=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 5 mō b, me -17 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 9\left(-17\right)}}{2\times 9}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 9\left(-17\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-5±\sqrt{25-36\left(-17\right)}}{2\times 9}$$

Whakareatia -36 ki te -17.

$$x=\frac{-5±\sqrt{25+612}}{2\times 9}$$

Tāpiri 25 ki te 612.

$$x=\frac{-5±\sqrt{637}}{2\times 9}$$

Tuhia te pūtakerua o te 637.

$$x=\frac{-5±7\sqrt{13}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-5±7\sqrt{13}}{18}$$

Nā, me whakaoti te whārite x=\frac{-5±7\sqrt{13}}{18} ina he tāpiri te ±. Tāpiri -5 ki te 7\sqrt{13}.

$$x=\frac{7\sqrt{13}-5}{18}$$

Nā, me whakaoti te whārite x=\frac{-5±7\sqrt{13}}{18} ina he tango te ±. Tango 7\sqrt{13} mai i -5.

$$x=\frac{-7\sqrt{13}-5}{18}$$

Kua oti te whārite te whakatau.

$$x=\frac{7\sqrt{13}-5}{18} x=\frac{-7\sqrt{13}-5}{18}$$

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.

$$9x^{2}+5x+3=20$$

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

$$9x^{2}+5x+3-3=20-3$$

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

$$9x^{2}+5x=20-3$$

Tango 3 mai i 20.

$$9x^{2}+5x=17$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+5x}{9}=\frac{17}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\frac{5}{9}x=\frac{17}{9}$$

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

$$x^{2}+\frac{5}{9}x+\left(\frac{5}{18}\right)^{2}=\frac{17}{9}+\left(\frac{5}{18}\right)^{2}$$

Pūruatia \frac{5}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=\frac{17}{9}+\frac{25}{324}$$

Tāpiri \frac{17}{9} ki te \frac{25}{324} 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.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=\frac{637}{324}$$

Tauwehea x^{2}+\frac{5}{9}x+\frac{25}{324}. 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}.

$$\left(x+\frac{5}{18}\right)^{2}=\frac{637}{324}$$

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

$$\sqrt{\left(x+\frac{5}{18}\right)^{2}}=\sqrt{\frac{637}{324}}$$

Whakarūnātia.

$$x+\frac{5}{18}=\frac{7\sqrt{13}}{18} x+\frac{5}{18}=-\frac{7\sqrt{13}}{18}$$

Me tango \frac{5}{18} mai i ngā taha e rua o te whārite.

$$x=\frac{7\sqrt{13}-5}{18} x=\frac{-7\sqrt{13}-5}{18}$$