Whakaoti i te (x+43)^2+(2x+34-8)^2=0

$$( x + 43 ) ^ { 2 } + ( 2 x + 34 - 8 ) ^ { 2 } = 0$$

Whakaoti mō x (complex solution)

$$x=-19+12i$$

$$x=-19-12i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+43\right)^{2}.

$$x^{2}+86x+1849+\left(2x+34-8\right)^{2}=0$$

Tangohia te 8 i te 34, ka 26.

$$x^{2}+86x+1849+\left(2x+26\right)^{2}=0$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+26\right)^{2}.

$$x^{2}+86x+1849+4x^{2}+104x+676=0$$

Pahekotia te x^{2} me 4x^{2}, ka 5x^{2}.

$$5x^{2}+86x+1849+104x+676=0$$

Pahekotia te 86x me 104x, ka 190x.

$$5x^{2}+190x+1849+676=0$$

Tāpirihia te 1849 ki te 676, ka 2525.

$$5x^{2}+190x+2525=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 190 mō b, me 2525 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-190±\sqrt{190^{2}-4\times 5\times 2525}}{2\times 5}$$

Pūrua 190.

$$x=\frac{-190±\sqrt{36100-4\times 5\times 2525}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-190±\sqrt{36100-20\times 2525}}{2\times 5}$$

Whakareatia -20 ki te 2525.

$$x=\frac{-190±\sqrt{36100-50500}}{2\times 5}$$

Tāpiri 36100 ki te -50500.

$$x=\frac{-190±\sqrt{-14400}}{2\times 5}$$

Tuhia te pūtakerua o te -14400.

$$x=\frac{-190±120i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-190±120i}{10}$$

Nā, me whakaoti te whārite x=\frac{-190±120i}{10} ina he tāpiri te ±. Tāpiri -190 ki te 120i.

$$x=\frac{-190+120i}{10}$$

Whakawehe -190+120i ki te 10.

$$x=-19+12i$$

Nā, me whakaoti te whārite x=\frac{-190±120i}{10} ina he tango te ±. Tango 120i mai i -190.

$$x=\frac{-190-120i}{10}$$

Whakawehe -190-120i ki te 10.

$$x=-19-12i$$

Kua oti te whārite te whakatau.

$$x=-19+12i x=-19-12i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+43\right)^{2}.

$$x^{2}+86x+1849+\left(2x+34-8\right)^{2}=0$$

Tangohia te 8 i te 34, ka 26.

$$x^{2}+86x+1849+\left(2x+26\right)^{2}=0$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+26\right)^{2}.

$$x^{2}+86x+1849+4x^{2}+104x+676=0$$

Pahekotia te x^{2} me 4x^{2}, ka 5x^{2}.

$$5x^{2}+86x+1849+104x+676=0$$

Pahekotia te 86x me 104x, ka 190x.

$$5x^{2}+190x+1849+676=0$$

Tāpirihia te 1849 ki te 676, ka 2525.

$$5x^{2}+190x+2525=0$$

Tangohia te 2525 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x^{2}+190x=-2525$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+190x}{5}=-\frac{2525}{5}$$

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

$$x^{2}+\frac{190}{5}x=-\frac{2525}{5}$$

Whakawehe 190 ki te 5.

$$x^{2}+38x=-\frac{2525}{5}$$

Whakawehe -2525 ki te 5.

$$x^{2}+38x=-505$$

Whakawehea te 38, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 19. Nā, tāpiria te pūrua o te 19 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+38x+19^{2}=-505+19^{2}$$

Pūrua 19.

$$x^{2}+38x+361=-505+361$$

Tāpiri -505 ki te 361.

$$x^{2}+38x+361=-144$$

Tauwehea x^{2}+38x+361. 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}.

$$\left(x+19\right)^{2}=-144$$

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

$$\sqrt{\left(x+19\right)^{2}}=\sqrt{-144}$$

Whakarūnātia.

$$x+19=12i x+19=-12i$$

Me tango 19 mai i ngā taha e rua o te whārite.

$$x=-19+12i x=-19-12i$$