Whakaoti i te 0=1/5left(x+5right)^2-1

$$0= \frac{ 1 }{ 5 } { \left(x+5 \right) }^{ 2 } -1$$

Whakaoti mō x

$$x=\sqrt{5}-5\approx -2.763932023$$

$$x=-\sqrt{5}-5\approx -7.236067977$$

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+5\right)^{2}.

$$0=\frac{1}{5}\left(x^{2}+10x+25\right)-1$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{5} ki te x^{2}+10x+25.

$$0=\frac{1}{5}x^{2}+2x+5-1$$

Tangohia te 1 i te 5, ka 4.

$$0=\frac{1}{5}x^{2}+2x+4$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$\frac{1}{5}x^{2}+2x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{5} mō a, 2 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times \frac{1}{5}\times 4}}{2\times \frac{1}{5}}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\times \frac{1}{5}\times 4}}{2\times \frac{1}{5}}$$

Whakareatia -4 ki te \frac{1}{5}.

$$x=\frac{-2±\sqrt{4-\frac{4}{5}\times 4}}{2\times \frac{1}{5}}$$

Whakareatia -\frac{4}{5} ki te 4.

$$x=\frac{-2±\sqrt{4-\frac{16}{5}}}{2\times \frac{1}{5}}$$

Tāpiri 4 ki te -\frac{16}{5}.

$$x=\frac{-2±\sqrt{\frac{4}{5}}}{2\times \frac{1}{5}}$$

Tuhia te pūtakerua o te \frac{4}{5}.

$$x=\frac{-2±\frac{2\sqrt{5}}{5}}{2\times \frac{1}{5}}$$

Whakareatia 2 ki te \frac{1}{5}.

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

Nā, me whakaoti te whārite x=\frac{-2±\frac{2\sqrt{5}}{5}}{\frac{2}{5}} ina he tāpiri te ±. Tāpiri -2 ki te \frac{2\sqrt{5}}{5}.

$$x=\frac{\frac{2\sqrt{5}}{5}-2}{\frac{2}{5}}$$

Whakawehe -2+\frac{2\sqrt{5}}{5} ki te \frac{2}{5} mā te whakarea -2+\frac{2\sqrt{5}}{5} ki te tau huripoki o \frac{2}{5}.

$$x=\sqrt{5}-5$$

Nā, me whakaoti te whārite x=\frac{-2±\frac{2\sqrt{5}}{5}}{\frac{2}{5}} ina he tango te ±. Tango \frac{2\sqrt{5}}{5} mai i -2.

$$x=\frac{-\frac{2\sqrt{5}}{5}-2}{\frac{2}{5}}$$

Whakawehe -2-\frac{2\sqrt{5}}{5} ki te \frac{2}{5} mā te whakarea -2-\frac{2\sqrt{5}}{5} ki te tau huripoki o \frac{2}{5}.

$$x=-\sqrt{5}-5$$

Kua oti te whārite te whakatau.

$$x=\sqrt{5}-5 x=-\sqrt{5}-5$$

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+5\right)^{2}.

$$0=\frac{1}{5}\left(x^{2}+10x+25\right)-1$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{5} ki te x^{2}+10x+25.

$$0=\frac{1}{5}x^{2}+2x+5-1$$

Tangohia te 1 i te 5, ka 4.

$$0=\frac{1}{5}x^{2}+2x+4$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$\frac{1}{5}x^{2}+2x+4=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$\frac{1}{5}x^{2}+2x=-4$$

Me whakarea ngā taha e rua ki te 5.

$$\frac{\frac{1}{5}x^{2}+2x}{\frac{1}{5}}=-\frac{4}{\frac{1}{5}}$$

Mā te whakawehe ki te \frac{1}{5} ka wetekia te whakareanga ki te \frac{1}{5}.

$$x^{2}+\frac{2}{\frac{1}{5}}x=-\frac{4}{\frac{1}{5}}$$

Whakawehe 2 ki te \frac{1}{5} mā te whakarea 2 ki te tau huripoki o \frac{1}{5}.

$$x^{2}+10x=-\frac{4}{\frac{1}{5}}$$

Whakawehe -4 ki te \frac{1}{5} mā te whakarea -4 ki te tau huripoki o \frac{1}{5}.

$$x^{2}+10x=-20$$

Whakawehea te 10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 5. Nā, tāpiria te pūrua o te 5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+10x+5^{2}=-20+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=-20+25$$

Tāpiri -20 ki te 25.

$$x^{2}+10x+25=5$$

Tauwehea te x^{2}+10x+25. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

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

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

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

Whakarūnātia.

$$x+5=\sqrt{5} x+5=-\sqrt{5}$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x=\sqrt{5}-5 x=-\sqrt{5}-5$$