Whakaoti i te 23x^2+5x+3=0

$$23 x ^ { 2 } + 5 x + 3 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-5+\sqrt{251}i}{46}\approx -0.108695652+0.344412598i$$

$$x=\frac{-\sqrt{251}i-5}{46}\approx -0.108695652-0.344412598i$$

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.

$$23x^{2}+5x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 23 mō a, 5 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 23\times 3}}{2\times 23}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 23\times 3}}{2\times 23}$$

Whakareatia -4 ki te 23.

$$x=\frac{-5±\sqrt{25-92\times 3}}{2\times 23}$$

Whakareatia -92 ki te 3.

$$x=\frac{-5±\sqrt{25-276}}{2\times 23}$$

Tāpiri 25 ki te -276.

$$x=\frac{-5±\sqrt{-251}}{2\times 23}$$

Tuhia te pūtakerua o te -251.

$$x=\frac{-5±\sqrt{251}i}{2\times 23}$$

Whakareatia 2 ki te 23.

$$x=\frac{-5±\sqrt{251}i}{46}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{251}i}{46} ina he tāpiri te ±. Tāpiri -5 ki te i\sqrt{251}.

$$x=\frac{-5+\sqrt{251}i}{46}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{251}i}{46} ina he tango te ±. Tango i\sqrt{251} mai i -5.

$$x=\frac{-\sqrt{251}i-5}{46}$$

Kua oti te whārite te whakatau.

$$x=\frac{-5+\sqrt{251}i}{46} x=\frac{-\sqrt{251}i-5}{46}$$

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.

$$23x^{2}+5x+3=0$$

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

$$23x^{2}+5x+3-3=-3$$

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

$$23x^{2}+5x=-3$$

Whakawehea ngā taha e rua ki te 23.

$$\frac{23x^{2}+5x}{23}=-\frac{3}{23}$$

Mā te whakawehe ki te 23 ka wetekia te whakareanga ki te 23.

$$x^{2}+\frac{5}{23}x=-\frac{3}{23}$$

Whakawehea te \frac{5}{23}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{46}. Nā, tāpiria te pūrua o te \frac{5}{46} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{23}x+\left(\frac{5}{46}\right)^{2}=-\frac{3}{23}+\left(\frac{5}{46}\right)^{2}$$

Pūruatia \frac{5}{46} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{23}x+\frac{25}{2116}=-\frac{3}{23}+\frac{25}{2116}$$

Tāpiri -\frac{3}{23} ki te \frac{25}{2116} 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.

$$x^{2}+\frac{5}{23}x+\frac{25}{2116}=-\frac{251}{2116}$$

Tauwehea x^{2}+\frac{5}{23}x+\frac{25}{2116}. 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}.

$$\left(x+\frac{5}{46}\right)^{2}=-\frac{251}{2116}$$

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

$$\sqrt{\left(x+\frac{5}{46}\right)^{2}}=\sqrt{-\frac{251}{2116}}$$

Whakarūnātia.

$$x+\frac{5}{46}=\frac{\sqrt{251}i}{46} x+\frac{5}{46}=-\frac{\sqrt{251}i}{46}$$

Me tango \frac{5}{46} mai i ngā taha e rua o te whārite.

$$x=\frac{-5+\sqrt{251}i}{46} x=\frac{-\sqrt{251}i-5}{46}$$