Whakaoti i te -4x^2+20x-47=0

$$-4 { x }^{ 2 } +20x-47=0$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{22}i+5}{2}\approx 2.5-2.34520788i$$

$$x=\frac{5+\sqrt{22}i}{2}\approx 2.5+2.34520788i$$

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.

$$-4x^{2}+20x-47=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, 20 mō b, me -47 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-4\right)\left(-47\right)}}{2\left(-4\right)}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\left(-4\right)\left(-47\right)}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$x=\frac{-20±\sqrt{400+16\left(-47\right)}}{2\left(-4\right)}$$

Whakareatia 16 ki te -47.

$$x=\frac{-20±\sqrt{400-752}}{2\left(-4\right)}$$

Tāpiri 400 ki te -752.

$$x=\frac{-20±\sqrt{-352}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te -352.

$$x=\frac{-20±4\sqrt{22}i}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$x=\frac{-20±4\sqrt{22}i}{-8}$$

Nā, me whakaoti te whārite x=\frac{-20±4\sqrt{22}i}{-8} ina he tāpiri te ±. Tāpiri -20 ki te 4i\sqrt{22}.

$$x=\frac{-20+4\sqrt{22}i}{-8}$$

Whakawehe -20+4i\sqrt{22} ki te -8.

$$x=\frac{-\sqrt{22}i+5}{2}$$

Nā, me whakaoti te whārite x=\frac{-20±4\sqrt{22}i}{-8} ina he tango te ±. Tango 4i\sqrt{22} mai i -20.

$$x=\frac{-4\sqrt{22}i-20}{-8}$$

Whakawehe -20-4i\sqrt{22} ki te -8.

$$x=\frac{5+\sqrt{22}i}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{22}i+5}{2} x=\frac{5+\sqrt{22}i}{2}$$

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.

$$-4x^{2}+20x-47=0$$

Me tāpiri 47 ki ngā taha e rua o te whārite.

$$-4x^{2}+20x-47-\left(-47\right)=-\left(-47\right)$$

Mā te tango i te -47 i a ia ake anō ka toe ko te 0.

$$-4x^{2}+20x=-\left(-47\right)$$

Tango -47 mai i 0.

$$-4x^{2}+20x=47$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4x^{2}+20x}{-4}=\frac{47}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$x^{2}+\frac{20}{-4}x=\frac{47}{-4}$$

Whakawehe 20 ki te -4.

$$x^{2}-5x=\frac{47}{-4}$$

Whakawehe 47 ki te -4.

$$x^{2}-5x=-\frac{47}{4}$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{47}{4}+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=\frac{-47+25}{4}$$

Tāpiri -\frac{47}{4} ki te \frac{25}{4} 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.

$$x^{2}-5x+\frac{25}{4}=-\frac{11}{2}$$

Tauwehea x^{2}-5x+\frac{25}{4}. 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}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{11}{2}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{11}{2}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{\sqrt{22}i}{2} x-\frac{5}{2}=-\frac{\sqrt{22}i}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=\frac{5+\sqrt{22}i}{2} x=\frac{-\sqrt{22}i+5}{2}$$