Whakaoti i te x(x-1)/3-x(x+1)/4+3x+4/12=0

$$\frac { x ( x - 1 ) } { 3 } - \frac { x ( x + 1 ) } { 4 } + \frac { 3 x + 4 } { 12 } = 0$$

Whakaoti mō x

$$x=2$$

Ngā Upane Whakamahi Tauwehe

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,12.

$$4x\left(x-1\right)-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x ki te x-1.

$$4x^{2}-4x-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3x ki te x+1.

$$4x^{2}-4x-3x^{2}-3x+3x+4=0$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}-4x-3x+3x+4=0$$

Pahekotia te -4x me -3x, ka -7x.

$$x^{2}-7x+3x+4=0$$

Pahekotia te -7x me 3x, ka -4x.

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

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

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

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 -4.

$$a=-2 b=-2$$

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

$$\left(x-2\right)\left(x-2\right)$$

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te x-2=0.

$$x=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,12.

$$4x\left(x-1\right)-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x ki te x-1.

$$4x^{2}-4x-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3x ki te x+1.

$$4x^{2}-4x-3x^{2}-3x+3x+4=0$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}-4x-3x+3x+4=0$$

Pahekotia te -4x me -3x, ka -7x.

$$x^{2}-7x+3x+4=0$$

Pahekotia te -7x me 3x, ka -4x.

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

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

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

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 -4.

$$a=-2 b=-2$$

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

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

Tauwehea te x i te tuatahi me te -2 i te rōpū tuarua.

$$x\left(x-2\right)-2\left(x-2\right)$$

Whakatauwehea atu te kīanga pātahi x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-2\right)\left(x-2\right)$$

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te x-2=0.

$$x=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,12.

$$4x\left(x-1\right)-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x ki te x-1.

$$4x^{2}-4x-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3x ki te x+1.

$$4x^{2}-4x-3x^{2}-3x+3x+4=0$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}-4x-3x+3x+4=0$$

Pahekotia te -4x me -3x, ka -7x.

$$x^{2}-7x+3x+4=0$$

Pahekotia te -7x me 3x, ka -4x.

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

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

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

Pūrua -4.

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

Whakareatia -4 ki te 4.

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

Tāpiri 16 ki te -16.

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

Tuhia te pūtakerua o te 0.

$$x=-\frac{-4}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,12.

$$4x\left(x-1\right)-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x ki te x-1.

$$4x^{2}-4x-3x\left(x+1\right)+3x+4=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3x ki te x+1.

$$4x^{2}-4x-3x^{2}-3x+3x+4=0$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}-4x-3x+3x+4=0$$

Pahekotia te -4x me -3x, ka -7x.

$$x^{2}-7x+3x+4=0$$

Pahekotia te -7x me 3x, ka -4x.

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

Tauwehea x^{2}-4x+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}.

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

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

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

Whakarūnātia.

$$x-2=0 x-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=2 x=2$$

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

$$x=2$$