Whakaoti i te 24x^2+46x+24=0

$$24 { x }^{ 2 } +46x+24=0$$

Whakaoti mō x (complex solution)

$$x=\frac{-23+\sqrt{47}i}{24}\approx -0.958333333+0.285652275i$$

$$x=\frac{-\sqrt{47}i-23}{24}\approx -0.958333333-0.285652275i$$

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.

$$24x^{2}+46x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 24 mō a, 46 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-46±\sqrt{46^{2}-4\times 24\times 24}}{2\times 24}$$

Pūrua 46.

$$x=\frac{-46±\sqrt{2116-4\times 24\times 24}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$x=\frac{-46±\sqrt{2116-96\times 24}}{2\times 24}$$

Whakareatia -96 ki te 24.

$$x=\frac{-46±\sqrt{2116-2304}}{2\times 24}$$

Tāpiri 2116 ki te -2304.

$$x=\frac{-46±\sqrt{-188}}{2\times 24}$$

Tuhia te pūtakerua o te -188.

$$x=\frac{-46±2\sqrt{47}i}{2\times 24}$$

Whakareatia 2 ki te 24.

$$x=\frac{-46±2\sqrt{47}i}{48}$$

Nā, me whakaoti te whārite x=\frac{-46±2\sqrt{47}i}{48} ina he tāpiri te ±. Tāpiri -46 ki te 2i\sqrt{47}.

$$x=\frac{-46+2\sqrt{47}i}{48}$$

Whakawehe -46+2i\sqrt{47} ki te 48.

$$x=\frac{-23+\sqrt{47}i}{24}$$

Nā, me whakaoti te whārite x=\frac{-46±2\sqrt{47}i}{48} ina he tango te ±. Tango 2i\sqrt{47} mai i -46.

$$x=\frac{-2\sqrt{47}i-46}{48}$$

Whakawehe -46-2i\sqrt{47} ki te 48.

$$x=\frac{-\sqrt{47}i-23}{24}$$

Kua oti te whārite te whakatau.

$$x=\frac{-23+\sqrt{47}i}{24} x=\frac{-\sqrt{47}i-23}{24}$$

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.

$$24x^{2}+46x+24=0$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$24x^{2}+46x+24-24=-24$$

Mā te tango i te 24 i a ia ake anō ka toe ko te 0.

$$24x^{2}+46x=-24$$

Whakawehea ngā taha e rua ki te 24.

$$\frac{24x^{2}+46x}{24}=-\frac{24}{24}$$

Mā te whakawehe ki te 24 ka wetekia te whakareanga ki te 24.

$$x^{2}+\frac{46}{24}x=-\frac{24}{24}$$

Whakahekea te hautanga \frac{46}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{23}{12}x=-\frac{24}{24}$$

Whakawehe -24 ki te 24.

$$x^{2}+\frac{23}{12}x=-1$$

Whakawehea te \frac{23}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{23}{24}. Nā, tāpiria te pūrua o te \frac{23}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{23}{12}x+\left(\frac{23}{24}\right)^{2}=-1+\left(\frac{23}{24}\right)^{2}$$

Pūruatia \frac{23}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{23}{12}x+\frac{529}{576}=-1+\frac{529}{576}$$

Tāpiri -1 ki te \frac{529}{576}.

$$x^{2}+\frac{23}{12}x+\frac{529}{576}=-\frac{47}{576}$$

Tauwehea x^{2}+\frac{23}{12}x+\frac{529}{576}. 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}.

$$\left(x+\frac{23}{24}\right)^{2}=-\frac{47}{576}$$

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

$$\sqrt{\left(x+\frac{23}{24}\right)^{2}}=\sqrt{-\frac{47}{576}}$$

Whakarūnātia.

$$x+\frac{23}{24}=\frac{\sqrt{47}i}{24} x+\frac{23}{24}=-\frac{\sqrt{47}i}{24}$$

Me tango \frac{23}{24} mai i ngā taha e rua o te whārite.

$$x=\frac{-23+\sqrt{47}i}{24} x=\frac{-\sqrt{47}i-23}{24}$$