Whakaoti i te r^2-5r+9=r

$$r ^ { 2 } - 5 r + 9 = r$$

Whakaoti mō r

$$r=3$$

Ngā Upane Whakamahi Tauwehe

Tangohia te r mai i ngā taha e rua.

$$r^{2}-5r+9-r=0$$

Pahekotia te -5r me -r, ka -6r.

$$r^{2}-6r+9=0$$

Hei whakaoti i te whārite, whakatauwehea te r^{2}-6r+9 mā te whakamahi i te tātai r^{2}+\left(a+b\right)r+ab=\left(r+a\right)\left(r+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=9$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

$$-1,-9 -3,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-3 b=-3$$

Me tuhi anō te kīanga whakatauwehe \left(r+a\right)\left(r+b\right) mā ngā uara i tātaihia.

$$\left(r-3\right)\left(r-3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(r-3\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te r-3=0.

$$r=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te r mai i ngā taha e rua.

$$r^{2}-5r+9-r=0$$

Pahekotia te -5r me -r, ka -6r.

$$r^{2}-6r+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei r^{2}+ar+br+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 9=9$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

$$-1,-9 -3,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-3 b=-3$$

Tuhia anō te r^{2}-6r+9 hei \left(r^{2}-3r\right)+\left(-3r+9\right).

$$\left(r^{2}-3r\right)+\left(-3r+9\right)$$

Tauwehea te r i te tuatahi me te -3 i te rōpū tuarua.

$$r\left(r-3\right)-3\left(r-3\right)$$

Whakatauwehea atu te kīanga pātahi r-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(r-3\right)\left(r-3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(r-3\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te r-3=0.

$$r=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te r mai i ngā taha e rua.

$$r^{2}-5r+9-r=0$$

Pahekotia te -5r me -r, ka -6r.

$$r^{2}-6r+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9}}{2}$$

Pūrua -6.

$$r=\frac{-\left(-6\right)±\sqrt{36-4\times 9}}{2}$$

Whakareatia -4 ki te 9.

$$r=\frac{-\left(-6\right)±\sqrt{36-36}}{2}$$

Tāpiri 36 ki te -36.

$$r=\frac{-\left(-6\right)±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

$$r=-\frac{-6}{2}$$

Ko te tauaro o -6 ko 6.

$$r=\frac{6}{2}$$

Whakawehe 6 ki te 2.

$$r=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te r mai i ngā taha e rua.

$$r^{2}-5r+9-r=0$$

Pahekotia te -5r me -r, ka -6r.

$$r^{2}-6r+9=0$$

Tauwehea r^{2}-6r+9. 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}.

$$\left(r-3\right)^{2}=0$$

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

$$\sqrt{\left(r-3\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$r-3=0 r-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$r=3 r=3$$

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

$$r=3$$