Whakaoti i te 7x^2+5x+5=0

$$7 x ^ { 2 } + 5 x + 5 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-5+\sqrt{115}i}{14}\approx -0.357142857+0.765986092i$$

$$x=\frac{-\sqrt{115}i-5}{14}\approx -0.357142857-0.765986092i$$

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.

$$7x^{2}+5x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 5 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 7\times 5}}{2\times 7}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 7\times 5}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-5±\sqrt{25-28\times 5}}{2\times 7}$$

Whakareatia -28 ki te 5.

$$x=\frac{-5±\sqrt{25-140}}{2\times 7}$$

Tāpiri 25 ki te -140.

$$x=\frac{-5±\sqrt{-115}}{2\times 7}$$

Tuhia te pūtakerua o te -115.

$$x=\frac{-5±\sqrt{115}i}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-5±\sqrt{115}i}{14}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{115}i}{14} ina he tāpiri te ±. Tāpiri -5 ki te i\sqrt{115}.

$$x=\frac{-5+\sqrt{115}i}{14}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{115}i}{14} ina he tango te ±. Tango i\sqrt{115} mai i -5.

$$x=\frac{-\sqrt{115}i-5}{14}$$

Kua oti te whārite te whakatau.

$$x=\frac{-5+\sqrt{115}i}{14} x=\frac{-\sqrt{115}i-5}{14}$$

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.

$$7x^{2}+5x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$7x^{2}+5x+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$7x^{2}+5x=-5$$

Whakawehea ngā taha e rua ki te 7.

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

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

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

Whakawehea te \frac{5}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{14}. Nā, tāpiria te pūrua o te \frac{5}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{7}x+\left(\frac{5}{14}\right)^{2}=-\frac{5}{7}+\left(\frac{5}{14}\right)^{2}$$

Pūruatia \frac{5}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=-\frac{5}{7}+\frac{25}{196}$$

Tāpiri -\frac{5}{7} ki te \frac{25}{196} 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.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=-\frac{115}{196}$$

Tauwehea x^{2}+\frac{5}{7}x+\frac{25}{196}. 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}.

$$\left(x+\frac{5}{14}\right)^{2}=-\frac{115}{196}$$

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

$$\sqrt{\left(x+\frac{5}{14}\right)^{2}}=\sqrt{-\frac{115}{196}}$$

Whakarūnātia.

$$x+\frac{5}{14}=\frac{\sqrt{115}i}{14} x+\frac{5}{14}=-\frac{\sqrt{115}i}{14}$$

Me tango \frac{5}{14} mai i ngā taha e rua o te whārite.

$$x=\frac{-5+\sqrt{115}i}{14} x=\frac{-\sqrt{115}i-5}{14}$$