Whakaoti i te -7x^2+5x-4=0

$$- 7 x ^ { 2 } + 5 x - 4 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{87}i+5}{14}\approx 0.357142857-0.666241361i$$

$$x=\frac{5+\sqrt{87}i}{14}\approx 0.357142857+0.666241361i$$

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.

$$-7x^{2}+5x-4=0$$

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

$$x=\frac{-5±\sqrt{5^{2}-4\left(-7\right)\left(-4\right)}}{2\left(-7\right)}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\left(-7\right)\left(-4\right)}}{2\left(-7\right)}$$

Whakareatia -4 ki te -7.

$$x=\frac{-5±\sqrt{25+28\left(-4\right)}}{2\left(-7\right)}$$

Whakareatia 28 ki te -4.

$$x=\frac{-5±\sqrt{25-112}}{2\left(-7\right)}$$

Tāpiri 25 ki te -112.

$$x=\frac{-5±\sqrt{-87}}{2\left(-7\right)}$$

Tuhia te pūtakerua o te -87.

$$x=\frac{-5±\sqrt{87}i}{2\left(-7\right)}$$

Whakareatia 2 ki te -7.

$$x=\frac{-5±\sqrt{87}i}{-14}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{87}i}{-14} ina he tāpiri te ±. Tāpiri -5 ki te i\sqrt{87}.

$$x=\frac{-5+\sqrt{87}i}{-14}$$

Whakawehe -5+i\sqrt{87} ki te -14.

$$x=\frac{-\sqrt{87}i+5}{14}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{87}i}{-14} ina he tango te ±. Tango i\sqrt{87} mai i -5.

$$x=\frac{-\sqrt{87}i-5}{-14}$$

Whakawehe -5-i\sqrt{87} ki te -14.

$$x=\frac{5+\sqrt{87}i}{14}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{87}i+5}{14} x=\frac{5+\sqrt{87}i}{14}$$

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.

$$-7x^{2}+5x-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$-7x^{2}+5x-4-\left(-4\right)=-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$-7x^{2}+5x=-\left(-4\right)$$

Tango -4 mai i 0.

$$-7x^{2}+5x=4$$

Whakawehea ngā taha e rua ki te -7.

$$\frac{-7x^{2}+5x}{-7}=\frac{4}{-7}$$

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

$$x^{2}+\frac{5}{-7}x=\frac{4}{-7}$$

Whakawehe 5 ki te -7.

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

Whakawehe 4 ki te -7.

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

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.

$$x^{2}-\frac{5}{7}x+\left(-\frac{5}{14}\right)^{2}=-\frac{4}{7}+\left(-\frac{5}{14}\right)^{2}$$

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

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=-\frac{4}{7}+\frac{25}{196}$$

Tāpiri -\frac{4}{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.

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=-\frac{87}{196}$$

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}.

$$\left(x-\frac{5}{14}\right)^{2}=-\frac{87}{196}$$

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

$$\sqrt{\left(x-\frac{5}{14}\right)^{2}}=\sqrt{-\frac{87}{196}}$$

Whakarūnātia.

$$x-\frac{5}{14}=\frac{\sqrt{87}i}{14} x-\frac{5}{14}=-\frac{\sqrt{87}i}{14}$$

Me tāpiri \frac{5}{14} ki ngā taha e rua o te whārite.

$$x=\frac{5+\sqrt{87}i}{14} x=\frac{-\sqrt{87}i+5}{14}$$