Whakaoti i te left(x+101right)^2+10(x+101)+9=0

$${ \left(x+101 \right) }^{ 2 } +10(x+101)+9=0$$

Whakaoti mō x

$$x=-110$$

$$x=-102$$

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+101\right)^{2}.

$$x^{2}+202x+10201+10\left(x+101\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+101.

$$x^{2}+202x+10201+10x+1010+9=0$$

Pahekotia te 202x me 10x, ka 212x.

$$x^{2}+212x+10201+1010+9=0$$

Tāpirihia te 10201 ki te 1010, ka 11211.

$$x^{2}+212x+11211+9=0$$

Tāpirihia te 11211 ki te 9, ka 11220.

$$x^{2}+212x+11220=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 212 mō b, me 11220 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-212±\sqrt{212^{2}-4\times 11220}}{2}$$

Pūrua 212.

$$x=\frac{-212±\sqrt{44944-4\times 11220}}{2}$$

Whakareatia -4 ki te 11220.

$$x=\frac{-212±\sqrt{44944-44880}}{2}$$

Tāpiri 44944 ki te -44880.

$$x=\frac{-212±\sqrt{64}}{2}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-212±8}{2}$$

Nā, me whakaoti te whārite x=\frac{-212±8}{2} ina he tāpiri te ±. Tāpiri -212 ki te 8.

$$x=-\frac{204}{2}$$

Whakawehe -204 ki te 2.

$$x=-102$$

Nā, me whakaoti te whārite x=\frac{-212±8}{2} ina he tango te ±. Tango 8 mai i -212.

$$x=-\frac{220}{2}$$

Whakawehe -220 ki te 2.

$$x=-110$$

Kua oti te whārite te whakatau.

$$x=-102 x=-110$$

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+101\right)^{2}.

$$x^{2}+202x+10201+10\left(x+101\right)+9=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+101.

$$x^{2}+202x+10201+10x+1010+9=0$$

Pahekotia te 202x me 10x, ka 212x.

$$x^{2}+212x+10201+1010+9=0$$

Tāpirihia te 10201 ki te 1010, ka 11211.

$$x^{2}+212x+11211+9=0$$

Tāpirihia te 11211 ki te 9, ka 11220.

$$x^{2}+212x+11220=0$$

Tangohia te 11220 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}+212x=-11220$$

Whakawehea te 212, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 106. Nā, tāpiria te pūrua o te 106 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+212x+106^{2}=-11220+106^{2}$$

Pūrua 106.

$$x^{2}+212x+11236=-11220+11236$$

Tāpiri -11220 ki te 11236.

$$x^{2}+212x+11236=16$$

Tauwehea x^{2}+212x+11236. 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}.

$$\left(x+106\right)^{2}=16$$

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

$$\sqrt{\left(x+106\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$x+106=4 x+106=-4$$

Me tango 106 mai i ngā taha e rua o te whārite.

$$x=-102 x=-110$$