Whakaoti i te r^2-2r^2+2=r^2+4-4r

$$r ^ { 2 } - 2 r ^ { 2 } + 2 = r ^ { 2 } + 4 - 4 r$$

Whakaoti mō r

$$r=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Pahekotia te r^{2} me -2r^{2}, ka -r^{2}.

$$-r^{2}+2=r^{2}+4-4r$$

Tangohia te r^{2} mai i ngā taha e rua.

$$-r^{2}+2-r^{2}=4-4r$$

Pahekotia te -r^{2} me -r^{2}, ka -2r^{2}.

$$-2r^{2}+2=4-4r$$

Tangohia te 4 mai i ngā taha e rua.

$$-2r^{2}+2-4=-4r$$

Tangohia te 4 i te 2, ka -2.

$$-2r^{2}-2=-4r$$

Me tāpiri te 4r ki ngā taha e rua.

$$-2r^{2}-2+4r=0$$

Whakawehea ngā taha e rua ki te 2.

$$-r^{2}-1+2r=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-r^{2}+2r-1=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -r^{2}+ar+br-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=-\left(-1\right)=1$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=1$$

Tuhia anō te -r^{2}+2r-1 hei \left(-r^{2}+r\right)+\left(r-1\right).

$$\left(-r^{2}+r\right)+\left(r-1\right)$$

Whakatauwehea atu -r i te -r^{2}+r.

$$-r\left(r-1\right)+r-1$$

Whakatauwehea atu te kīanga pātahi r-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(r-1\right)\left(-r+1\right)$$

Hei kimi otinga whārite, me whakaoti te r-1=0 me te -r+1=0.

$$r=1 r=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te r^{2} me -2r^{2}, ka -r^{2}.

$$-r^{2}+2=r^{2}+4-4r$$

Tangohia te r^{2} mai i ngā taha e rua.

$$-r^{2}+2-r^{2}=4-4r$$

Pahekotia te -r^{2} me -r^{2}, ka -2r^{2}.

$$-2r^{2}+2=4-4r$$

Tangohia te 4 mai i ngā taha e rua.

$$-2r^{2}+2-4=-4r$$

Tangohia te 4 i te 2, ka -2.

$$-2r^{2}-2=-4r$$

Me tāpiri te 4r ki ngā taha e rua.

$$-2r^{2}-2+4r=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-2r^{2}+4r-2=0$$

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

$$r=\frac{-4±\sqrt{4^{2}-4\left(-2\right)\left(-2\right)}}{2\left(-2\right)}$$

Pūrua 4.

$$r=\frac{-4±\sqrt{16-4\left(-2\right)\left(-2\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$r=\frac{-4±\sqrt{16+8\left(-2\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -2.

$$r=\frac{-4±\sqrt{16-16}}{2\left(-2\right)}$$

Tāpiri 16 ki te -16.

$$r=\frac{-4±\sqrt{0}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 0.

$$r=-\frac{4}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$r=-\frac{4}{-4}$$

Whakawehe -4 ki te -4.

$$r=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te r^{2} me -2r^{2}, ka -r^{2}.

$$-r^{2}+2=r^{2}+4-4r$$

Tangohia te r^{2} mai i ngā taha e rua.

$$-r^{2}+2-r^{2}=4-4r$$

Pahekotia te -r^{2} me -r^{2}, ka -2r^{2}.

$$-2r^{2}+2=4-4r$$

Me tāpiri te 4r ki ngā taha e rua.

$$-2r^{2}+2+4r=4$$

Tangohia te 2 mai i ngā taha e rua.

$$-2r^{2}+4r=4-2$$

Tangohia te 2 i te 4, ka 2.

$$-2r^{2}+4r=2$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2r^{2}+4r}{-2}=\frac{2}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$r^{2}+\frac{4}{-2}r=\frac{2}{-2}$$

Whakawehe 4 ki te -2.

$$r^{2}-2r=\frac{2}{-2}$$

Whakawehe 2 ki te -2.

$$r^{2}-2r=-1$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}-2r+1=-1+1$$

Tāpiri -1 ki te 1.

$$r^{2}-2r+1=0$$

Tauwehea r^{2}-2r+1. 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}.

$$\left(r-1\right)^{2}=0$$

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

$$\sqrt{\left(r-1\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$r-1=0 r-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$r=1 r=1$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$r=1$$