Whakaoti i te (x-3)^2=x/4

$$( x - 3 ) ^ { 2 } = \frac { x } { 4 }$$

Whakaoti mō x

$$x=4$$

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4\left(x-3\right)^{2}=x$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-3\right)^{2}.

$$4\left(x^{2}-6x+9\right)=x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}-6x+9.

$$4x^{2}-24x+36=x$$

Tangohia te x mai i ngā taha e rua.

$$4x^{2}-24x+36-x=0$$

Pahekotia te -24x me -x, ka -25x.

$$4x^{2}-25x+36=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-25 ab=4\times 36=144$$

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 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

Ko te otinga te takirua ka hoatu i te tapeke -25.

$$a=-16 b=-9$$

Tuhia anō te 4x^{2}-25x+36 hei \left(4x^{2}-16x\right)+\left(-9x+36\right).

$$\left(4x^{2}-16x\right)+\left(-9x+36\right)$$

Tauwehea te 4x i te tuatahi me te -9 i te rōpū tuarua.

$$4x\left(x-4\right)-9\left(x-4\right)$$

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

$$\left(x-4\right)\left(4x-9\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te 4x-9=0.

$$x=4 x=\frac{9}{4}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4\left(x-3\right)^{2}=x$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-3\right)^{2}.

$$4\left(x^{2}-6x+9\right)=x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}-6x+9.

$$4x^{2}-24x+36=x$$

Tangohia te x mai i ngā taha e rua.

$$4x^{2}-24x+36-x=0$$

Pahekotia te -24x me -x, ka -25x.

$$4x^{2}-25x+36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -25 mō b, me 36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 4\times 36}}{2\times 4}$$

Pūrua -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 4\times 36}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-25\right)±\sqrt{625-16\times 36}}{2\times 4}$$

Whakareatia -16 ki te 36.

$$x=\frac{-\left(-25\right)±\sqrt{625-576}}{2\times 4}$$

Tāpiri 625 ki te -576.

$$x=\frac{-\left(-25\right)±\sqrt{49}}{2\times 4}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-25\right)±7}{2\times 4}$$

Ko te tauaro o -25 ko 25.

$$x=\frac{25±7}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{25±7}{8}$$

Nā, me whakaoti te whārite x=\frac{25±7}{8} ina he tāpiri te ±. Tāpiri 25 ki te 7.

$$x=\frac{32}{8}$$

Whakawehe 32 ki te 8.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{25±7}{8} ina he tango te ±. Tango 7 mai i 25.

$$x=\frac{18}{8}$$

Whakahekea te hautanga \frac{18}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{9}{4}$$

Kua oti te whārite te whakatau.

$$x=4 x=\frac{9}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4\left(x-3\right)^{2}=x$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-3\right)^{2}.

$$4\left(x^{2}-6x+9\right)=x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}-6x+9.

$$4x^{2}-24x+36=x$$

Tangohia te x mai i ngā taha e rua.

$$4x^{2}-24x+36-x=0$$

Pahekotia te -24x me -x, ka -25x.

$$4x^{2}-25x+36=0$$

Tangohia te 36 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4x^{2}-25x=-36$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-25x}{4}=-\frac{36}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}-\frac{25}{4}x=-\frac{36}{4}$$

Whakawehe -36 ki te 4.

$$x^{2}-\frac{25}{4}x=-9$$

Whakawehea te -\frac{25}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{25}{8}. Nā, tāpiria te pūrua o te -\frac{25}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{25}{4}x+\left(-\frac{25}{8}\right)^{2}=-9+\left(-\frac{25}{8}\right)^{2}$$

Pūruatia -\frac{25}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=-9+\frac{625}{64}$$

Tāpiri -9 ki te \frac{625}{64}.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=\frac{49}{64}$$

Tauwehea x^{2}-\frac{25}{4}x+\frac{625}{64}. 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}.

$$\left(x-\frac{25}{8}\right)^{2}=\frac{49}{64}$$

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

$$\sqrt{\left(x-\frac{25}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Whakarūnātia.

$$x-\frac{25}{8}=\frac{7}{8} x-\frac{25}{8}=-\frac{7}{8}$$

Me tāpiri \frac{25}{8} ki ngā taha e rua o te whārite.

$$x=4 x=\frac{9}{4}$$