Whakaoti i te 5x(x+6)=-50

$$5 x ( x + 6 ) = - 50$$

Whakaoti mō x (complex solution)

$$x=-3+i$$

$$x=-3-i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x+6.

$$5x^{2}+30x=-50$$

Me tāpiri te 50 ki ngā taha e rua.

$$5x^{2}+30x+50=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 30 mō b, me 50 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 5\times 50}}{2\times 5}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\times 5\times 50}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-30±\sqrt{900-20\times 50}}{2\times 5}$$

Whakareatia -20 ki te 50.

$$x=\frac{-30±\sqrt{900-1000}}{2\times 5}$$

Tāpiri 900 ki te -1000.

$$x=\frac{-30±\sqrt{-100}}{2\times 5}$$

Tuhia te pūtakerua o te -100.

$$x=\frac{-30±10i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-30±10i}{10}$$

Nā, me whakaoti te whārite x=\frac{-30±10i}{10} ina he tāpiri te ±. Tāpiri -30 ki te 10i.

$$x=\frac{-30+10i}{10}$$

Whakawehe -30+10i ki te 10.

$$x=-3+i$$

Nā, me whakaoti te whārite x=\frac{-30±10i}{10} ina he tango te ±. Tango 10i mai i -30.

$$x=\frac{-30-10i}{10}$$

Whakawehe -30-10i ki te 10.

$$x=-3-i$$

Kua oti te whārite te whakatau.

$$x=-3+i x=-3-i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x+6.

$$5x^{2}+30x=-50$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+30x}{5}=-\frac{50}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\frac{30}{5}x=-\frac{50}{5}$$

Whakawehe 30 ki te 5.

$$x^{2}+6x=-\frac{50}{5}$$

Whakawehe -50 ki te 5.

$$x^{2}+6x=-10$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+6x+3^{2}=-10+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=-10+9$$

Tāpiri -10 ki te 9.

$$x^{2}+6x+9=-1$$

Tauwehea x^{2}+6x+9. 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}.

$$\left(x+3\right)^{2}=-1$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{-1}$$

Whakarūnātia.

$$x+3=i x+3=-i$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=-3+i x=-3-i$$