Whakaoti i te (y-1)^2-3(y-1)=0

$$( y - 1 ) ^ { 2 } - 3 ( y - 1 ) = 0$$

Whakaoti mō y

$$y=4$$

$$y=1$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(y-1\right)^{2}.

$$y^{2}-2y+1-3\left(y-1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te y-1.

$$y^{2}-2y+1-3y+3=0$$

Pahekotia te -2y me -3y, ka -5y.

$$y^{2}-5y+1+3=0$$

Tāpirihia te 1 ki te 3, ka 4.

$$y^{2}-5y+4=0$$

Hei whakaoti i te whārite, whakatauwehea te y^{2}-5y+4 mā te whakamahi i te tātai y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=4$$

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 4.

$$-1,-4 -2,-2$$

Tātaihia te tapeke mō ia takirua.

$$-1-4=-5 -2-2=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-4 b=-1$$

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

$$\left(y-4\right)\left(y-1\right)$$

Hei kimi otinga whārite, me whakaoti te y-4=0 me te y-1=0.

$$y=4 y=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(y-1\right)^{2}.

$$y^{2}-2y+1-3\left(y-1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te y-1.

$$y^{2}-2y+1-3y+3=0$$

Pahekotia te -2y me -3y, ka -5y.

$$y^{2}-5y+1+3=0$$

Tāpirihia te 1 ki te 3, ka 4.

$$y^{2}-5y+4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei y^{2}+ay+by+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=1\times 4=4$$

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 4.

$$-1,-4 -2,-2$$

Tātaihia te tapeke mō ia takirua.

$$-1-4=-5 -2-2=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-4 b=-1$$

Tuhia anō te y^{2}-5y+4 hei \left(y^{2}-4y\right)+\left(-y+4\right).

$$\left(y^{2}-4y\right)+\left(-y+4\right)$$

Tauwehea te y i te tuatahi me te -1 i te rōpū tuarua.

$$y\left(y-4\right)-\left(y-4\right)$$

Whakatauwehea atu te kīanga pātahi y-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-4\right)\left(y-1\right)$$

Hei kimi otinga whārite, me whakaoti te y-4=0 me te y-1=0.

$$y=4 y=1$$

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(y-1\right)^{2}.

$$y^{2}-2y+1-3\left(y-1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te y-1.

$$y^{2}-2y+1-3y+3=0$$

Pahekotia te -2y me -3y, ka -5y.

$$y^{2}-5y+1+3=0$$

Tāpirihia te 1 ki te 3, ka 4.

$$y^{2}-5y+4=0$$

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

$$y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4}}{2}$$

Pūrua -5.

$$y=\frac{-\left(-5\right)±\sqrt{25-4\times 4}}{2}$$

Whakareatia -4 ki te 4.

$$y=\frac{-\left(-5\right)±\sqrt{25-16}}{2}$$

Tāpiri 25 ki te -16.

$$y=\frac{-\left(-5\right)±\sqrt{9}}{2}$$

Tuhia te pūtakerua o te 9.

$$y=\frac{-\left(-5\right)±3}{2}$$

Ko te tauaro o -5 ko 5.

$$y=\frac{5±3}{2}$$

Nā, me whakaoti te whārite y=\frac{5±3}{2} ina he tāpiri te ±. Tāpiri 5 ki te 3.

$$y=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$y=4$$

Nā, me whakaoti te whārite y=\frac{5±3}{2} ina he tango te ±. Tango 3 mai i 5.

$$y=\frac{2}{2}$$

Whakawehe 2 ki te 2.

$$y=1$$

Kua oti te whārite te whakatau.

$$y=4 y=1$$

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(y-1\right)^{2}.

$$y^{2}-2y+1-3\left(y-1\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te y-1.

$$y^{2}-2y+1-3y+3=0$$

Pahekotia te -2y me -3y, ka -5y.

$$y^{2}-5y+1+3=0$$

Tāpirihia te 1 ki te 3, ka 4.

$$y^{2}-5y+4=0$$

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

$$y^{2}-5y=-4$$

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

$$y^{2}-5y+\left(-\frac{5}{2}\right)^{2}=-4+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-5y+\frac{25}{4}=-4+\frac{25}{4}$$

Tāpiri -4 ki te \frac{25}{4}.

$$y^{2}-5y+\frac{25}{4}=\frac{9}{4}$$

Tauwehea y^{2}-5y+\frac{25}{4}. 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}.

$$\left(y-\frac{5}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(y-\frac{5}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$y-\frac{5}{2}=\frac{3}{2} y-\frac{5}{2}=-\frac{3}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$y=4 y=1$$