Whakaoti i te x^2-10x=-39

$$x ^ { 2 } - 10 x = - 39$$

Whakaoti mō x (complex solution)

$$x=5+\sqrt{14}i\approx 5+3.741657387i$$

$$x=-\sqrt{14}i+5\approx 5-3.741657387i$$

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.

$$x^{2}-10x=-39$$

Me tāpiri 39 ki ngā taha e rua o te whārite.

$$x^{2}-10x-\left(-39\right)=-39-\left(-39\right)$$

Mā te tango i te -39 i a ia ake anō ka toe ko te 0.

$$x^{2}-10x-\left(-39\right)=0$$

Tango -39 mai i 0.

$$x^{2}-10x+39=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me 39 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 39}}{2}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 39}}{2}$$

Whakareatia -4 ki te 39.

$$x=\frac{-\left(-10\right)±\sqrt{100-156}}{2}$$

Tāpiri 100 ki te -156.

$$x=\frac{-\left(-10\right)±\sqrt{-56}}{2}$$

Tuhia te pūtakerua o te -56.

$$x=\frac{-\left(-10\right)±2\sqrt{14}i}{2}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±2\sqrt{14}i}{2}$$

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{14}i}{2} ina he tāpiri te ±. Tāpiri 10 ki te 2i\sqrt{14}.

$$x=\frac{10+2\sqrt{14}i}{2}$$

Whakawehe 10+2i\sqrt{14} ki te 2.

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

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{14}i}{2} ina he tango te ±. Tango 2i\sqrt{14} mai i 10.

$$x=\frac{-2\sqrt{14}i+10}{2}$$

Whakawehe 10-2i\sqrt{14} ki te 2.

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

Kua oti te whārite te whakatau.

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

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.

$$x^{2}-10x=-39$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-10x+\left(-5\right)^{2}=-39+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=-39+25$$

Tāpiri -39 ki te 25.

$$x^{2}-10x+25=-14$$

Tauwehea x^{2}-10x+25. 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}.

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

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

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

Whakarūnātia.

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

Me tāpiri 5 ki ngā taha e rua o te whārite.

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