Whakaoti i te x^2-4x+8+x^2/2-14x+100=-x/2-2x+52

$${ x }^{ 2 } -4x+8+ \frac{ { x }^{ 2 } }{ 2 } -14x+100 = - \frac{ x }{ 2 } -2x+52$$

Whakaoti mō x (complex solution)

$$x=\frac{31+\sqrt{383}i}{6}\approx 5.166666667+3.261730965i$$

$$x=\frac{-\sqrt{383}i+31}{6}\approx 5.166666667-3.261730965i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x^{2}-8x+16+x^{2}-28x+200=-x-4x+104$$

Pahekotia te 2x^{2} me x^{2}, ka 3x^{2}.

$$3x^{2}-8x+16-28x+200=-x-4x+104$$

Pahekotia te -8x me -28x, ka -36x.

$$3x^{2}-36x+16+200=-x-4x+104$$

Tāpirihia te 16 ki te 200, ka 216.

$$3x^{2}-36x+216=-x-4x+104$$

Me tāpiri te x ki ngā taha e rua.

$$3x^{2}-36x+216+x=-4x+104$$

Pahekotia te -36x me x, ka -35x.

$$3x^{2}-35x+216=-4x+104$$

Me tāpiri te 4x ki ngā taha e rua.

$$3x^{2}-35x+216+4x=104$$

Pahekotia te -35x me 4x, ka -31x.

$$3x^{2}-31x+216=104$$

Tangohia te 104 mai i ngā taha e rua.

$$3x^{2}-31x+216-104=0$$

Tangohia te 104 i te 216, ka 112.

$$3x^{2}-31x+112=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -31 mō b, me 112 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 3\times 112}}{2\times 3}$$

Pūrua -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 3\times 112}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-31\right)±\sqrt{961-12\times 112}}{2\times 3}$$

Whakareatia -12 ki te 112.

$$x=\frac{-\left(-31\right)±\sqrt{961-1344}}{2\times 3}$$

Tāpiri 961 ki te -1344.

$$x=\frac{-\left(-31\right)±\sqrt{-383}}{2\times 3}$$

Tuhia te pūtakerua o te -383.

$$x=\frac{-\left(-31\right)±\sqrt{383}i}{2\times 3}$$

Ko te tauaro o -31 ko 31.

$$x=\frac{31±\sqrt{383}i}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{31±\sqrt{383}i}{6}$$

Nā, me whakaoti te whārite x=\frac{31±\sqrt{383}i}{6} ina he tāpiri te ±. Tāpiri 31 ki te i\sqrt{383}.

$$x=\frac{31+\sqrt{383}i}{6}$$

Nā, me whakaoti te whārite x=\frac{31±\sqrt{383}i}{6} ina he tango te ±. Tango i\sqrt{383} mai i 31.

$$x=\frac{-\sqrt{383}i+31}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{31+\sqrt{383}i}{6} x=\frac{-\sqrt{383}i+31}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x^{2}-8x+16+x^{2}-28x+200=-x-4x+104$$

Pahekotia te 2x^{2} me x^{2}, ka 3x^{2}.

$$3x^{2}-8x+16-28x+200=-x-4x+104$$

Pahekotia te -8x me -28x, ka -36x.

$$3x^{2}-36x+16+200=-x-4x+104$$

Tāpirihia te 16 ki te 200, ka 216.

$$3x^{2}-36x+216=-x-4x+104$$

Me tāpiri te x ki ngā taha e rua.

$$3x^{2}-36x+216+x=-4x+104$$

Pahekotia te -36x me x, ka -35x.

$$3x^{2}-35x+216=-4x+104$$

Me tāpiri te 4x ki ngā taha e rua.

$$3x^{2}-35x+216+4x=104$$

Pahekotia te -35x me 4x, ka -31x.

$$3x^{2}-31x+216=104$$

Tangohia te 216 mai i ngā taha e rua.

$$3x^{2}-31x=104-216$$

Tangohia te 216 i te 104, ka -112.

$$3x^{2}-31x=-112$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-31x}{3}=-\frac{112}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{31}{3}x=-\frac{112}{3}$$

Whakawehea te -\frac{31}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{31}{6}. Nā, tāpiria te pūrua o te -\frac{31}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{31}{3}x+\left(-\frac{31}{6}\right)^{2}=-\frac{112}{3}+\left(-\frac{31}{6}\right)^{2}$$

Pūruatia -\frac{31}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=-\frac{112}{3}+\frac{961}{36}$$

Tāpiri -\frac{112}{3} ki te \frac{961}{36} 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.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=-\frac{383}{36}$$

Tauwehea x^{2}-\frac{31}{3}x+\frac{961}{36}. 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}.

$$\left(x-\frac{31}{6}\right)^{2}=-\frac{383}{36}$$

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

$$\sqrt{\left(x-\frac{31}{6}\right)^{2}}=\sqrt{-\frac{383}{36}}$$

Whakarūnātia.

$$x-\frac{31}{6}=\frac{\sqrt{383}i}{6} x-\frac{31}{6}=-\frac{\sqrt{383}i}{6}$$

Me tāpiri \frac{31}{6} ki ngā taha e rua o te whārite.

$$x=\frac{31+\sqrt{383}i}{6} x=\frac{-\sqrt{383}i+31}{6}$$