Whakaoti i te (12-2x+1)x=80

$$(12-2x+1)x=80$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{471}i+13}{4}\approx 3.25-5.425633604i$$

$$x=\frac{13+\sqrt{471}i}{4}\approx 3.25+5.425633604i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tāpirihia te 12 ki te 1, ka 13.

$$\left(13-2x\right)x=80$$

Whakamahia te āhuatanga tohatoha hei whakarea te 13-2x ki te x.

$$13x-2x^{2}=80$$

Tangohia te 80 mai i ngā taha e rua.

$$13x-2x^{2}-80=0$$

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.

$$-2x^{2}+13x-80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 13 mō b, me -80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\left(-80\right)}}{2\left(-2\right)}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\left(-80\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-13±\sqrt{169+8\left(-80\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -80.

$$x=\frac{-13±\sqrt{169-640}}{2\left(-2\right)}$$

Tāpiri 169 ki te -640.

$$x=\frac{-13±\sqrt{-471}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te -471.

$$x=\frac{-13±\sqrt{471}i}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-13±\sqrt{471}i}{-4}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{471}i}{-4} ina he tāpiri te ±. Tāpiri -13 ki te i\sqrt{471}.

$$x=\frac{-13+\sqrt{471}i}{-4}$$

Whakawehe -13+i\sqrt{471} ki te -4.

$$x=\frac{-\sqrt{471}i+13}{4}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{471}i}{-4} ina he tango te ±. Tango i\sqrt{471} mai i -13.

$$x=\frac{-\sqrt{471}i-13}{-4}$$

Whakawehe -13-i\sqrt{471} ki te -4.

$$x=\frac{13+\sqrt{471}i}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{471}i+13}{4} x=\frac{13+\sqrt{471}i}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tāpirihia te 12 ki te 1, ka 13.

$$\left(13-2x\right)x=80$$

Whakamahia te āhuatanga tohatoha hei whakarea te 13-2x ki te x.

$$13x-2x^{2}=80$$

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.

$$-2x^{2}+13x=80$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+13x}{-2}=\frac{80}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$x^{2}+\frac{13}{-2}x=\frac{80}{-2}$$

Whakawehe 13 ki te -2.

$$x^{2}-\frac{13}{2}x=\frac{80}{-2}$$

Whakawehe 80 ki te -2.

$$x^{2}-\frac{13}{2}x=-40$$

Whakawehea te -\frac{13}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{4}. Nā, tāpiria te pūrua o te -\frac{13}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-40+\left(-\frac{13}{4}\right)^{2}$$

Pūruatia -\frac{13}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-40+\frac{169}{16}$$

Tāpiri -40 ki te \frac{169}{16}.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{471}{16}$$

Tauwehea x^{2}-\frac{13}{2}x+\frac{169}{16}. 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}.

$$\left(x-\frac{13}{4}\right)^{2}=-\frac{471}{16}$$

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

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{-\frac{471}{16}}$$

Whakarūnātia.

$$x-\frac{13}{4}=\frac{\sqrt{471}i}{4} x-\frac{13}{4}=-\frac{\sqrt{471}i}{4}$$

Me tāpiri \frac{13}{4} ki ngā taha e rua o te whārite.

$$x=\frac{13+\sqrt{471}i}{4} x=\frac{-\sqrt{471}i+13}{4}$$