Whakaoti i te (x-5)^2/2+x=3

$$\frac { ( x - 5 ) ^ { 2 } } { 2 } + x = 3$$

Whakaoti mō x (complex solution)

$$x=4+\sqrt{3}i\approx 4+1.732050808i$$

$$x=-\sqrt{3}i+4\approx 4-1.732050808i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$\left(x-5\right)^{2}+2x=6$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-5\right)^{2}.

$$x^{2}-10x+25+2x=6$$

Pahekotia te -10x me 2x, ka -8x.

$$x^{2}-8x+25=6$$

Tangohia te 6 mai i ngā taha e rua.

$$x^{2}-8x+25-6=0$$

Tangohia te 6 i te 25, ka 19.

$$x^{2}-8x+19=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 19 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 19}}{2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 19}}{2}$$

Whakareatia -4 ki te 19.

$$x=\frac{-\left(-8\right)±\sqrt{64-76}}{2}$$

Tāpiri 64 ki te -76.

$$x=\frac{-\left(-8\right)±\sqrt{-12}}{2}$$

Tuhia te pūtakerua o te -12.

$$x=\frac{-\left(-8\right)±2\sqrt{3}i}{2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2\sqrt{3}i}{2}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{3}i}{2} ina he tāpiri te ±. Tāpiri 8 ki te 2i\sqrt{3}.

$$x=\frac{8+2\sqrt{3}i}{2}$$

Whakawehe 8+2i\sqrt{3} ki te 2.

$$x=4+\sqrt{3}i$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{3}i}{2} ina he tango te ±. Tango 2i\sqrt{3} mai i 8.

$$x=\frac{-2\sqrt{3}i+8}{2}$$

Whakawehe 8-2i\sqrt{3} ki te 2.

$$x=-\sqrt{3}i+4$$

Kua oti te whārite te whakatau.

$$x=4+\sqrt{3}i x=-\sqrt{3}i+4$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$\left(x-5\right)^{2}+2x=6$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-5\right)^{2}.

$$x^{2}-10x+25+2x=6$$

Pahekotia te -10x me 2x, ka -8x.

$$x^{2}-8x+25=6$$

Tangohia te 25 mai i ngā taha e rua.

$$x^{2}-8x=6-25$$

Tangohia te 25 i te 6, ka -19.

$$x^{2}-8x=-19$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=-19+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-19+16$$

Tāpiri -19 ki te 16.

$$x^{2}-8x+16=-3$$

Tauwehea x^{2}-8x+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}.

$$\left(x-4\right)^{2}=-3$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-3}$$

Whakarūnātia.

$$x-4=\sqrt{3}i x-4=-\sqrt{3}i$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=4+\sqrt{3}i x=-\sqrt{3}i+4$$