Whakaoti i te 2x^2-7x+9=(x-3)(x+1)+3x

$$2 x ^ { 2 } - 7 x + 9 = ( x - 3 ) ( x + 1 ) + 3 x$$

Whakaoti mō x

$$x=2$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}-7x+9=x^{2}-2x-3+3x$$

Pahekotia te -2x me 3x, ka x.

$$2x^{2}-7x+9=x^{2}+x-3$$

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

$$2x^{2}-7x+9-x^{2}=x-3$$

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

$$x^{2}-7x+9=x-3$$

Tangohia te x mai i ngā taha e rua.

$$x^{2}-7x+9-x=-3$$

Pahekotia te -7x me -x, ka -8x.

$$x^{2}-8x+9=-3$$

Me tāpiri te 3 ki ngā taha e rua.

$$x^{2}-8x+9+3=0$$

Tāpirihia te 9 ki te 3, ka 12.

$$x^{2}-8x+12=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-8x+12 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.

$$a+b=-8 ab=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-6 b=-2$$

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

$$\left(x-6\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x-2=0.

$$x=6 x=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}-7x+9=x^{2}-2x-3+3x$$

Pahekotia te -2x me 3x, ka x.

$$2x^{2}-7x+9=x^{2}+x-3$$

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

$$2x^{2}-7x+9-x^{2}=x-3$$

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

$$x^{2}-7x+9=x-3$$

Tangohia te x mai i ngā taha e rua.

$$x^{2}-7x+9-x=-3$$

Pahekotia te -7x me -x, ka -8x.

$$x^{2}-8x+9=-3$$

Me tāpiri te 3 ki ngā taha e rua.

$$x^{2}-8x+9+3=0$$

Tāpirihia te 9 ki te 3, ka 12.

$$x^{2}-8x+12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\times 12=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-6 b=-2$$

Tuhia anō te x^{2}-8x+12 hei \left(x^{2}-6x\right)+\left(-2x+12\right).

$$\left(x^{2}-6x\right)+\left(-2x+12\right)$$

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

$$x\left(x-6\right)-2\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x-2=0.

$$x=6 x=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}-7x+9=x^{2}-2x-3+3x$$

Pahekotia te -2x me 3x, ka x.

$$2x^{2}-7x+9=x^{2}+x-3$$

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

$$2x^{2}-7x+9-x^{2}=x-3$$

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

$$x^{2}-7x+9=x-3$$

Tangohia te x mai i ngā taha e rua.

$$x^{2}-7x+9-x=-3$$

Pahekotia te -7x me -x, ka -8x.

$$x^{2}-8x+9=-3$$

Me tāpiri te 3 ki ngā taha e rua.

$$x^{2}-8x+9+3=0$$

Tāpirihia te 9 ki te 3, ka 12.

$$x^{2}-8x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 12}}{2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 12}}{2}$$

Whakareatia -4 ki te 12.

$$x=\frac{-\left(-8\right)±\sqrt{64-48}}{2}$$

Tāpiri 64 ki te -48.

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

Tuhia te pūtakerua o te 16.

$$x=\frac{-\left(-8\right)±4}{2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±4}{2}$$

Nā, me whakaoti te whārite x=\frac{8±4}{2} ina he tāpiri te ±. Tāpiri 8 ki te 4.

$$x=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{8±4}{2} ina he tango te ±. Tango 4 mai i 8.

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

Whakawehe 4 ki te 2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=6 x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}-7x+9=x^{2}-2x-3+3x$$

Pahekotia te -2x me 3x, ka x.

$$2x^{2}-7x+9=x^{2}+x-3$$

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

$$2x^{2}-7x+9-x^{2}=x-3$$

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

$$x^{2}-7x+9=x-3$$

Tangohia te x mai i ngā taha e rua.

$$x^{2}-7x+9-x=-3$$

Pahekotia te -7x me -x, ka -8x.

$$x^{2}-8x+9=-3$$

Tangohia te 9 mai i ngā taha e rua.

$$x^{2}-8x=-3-9$$

Tangohia te 9 i te -3, ka -12.

$$x^{2}-8x=-12$$

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

$$x^{2}-8x+\left(-4\right)^{2}=-12+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-12+16$$

Tāpiri -12 ki te 16.

$$x^{2}-8x+16=4$$

Tauwehea x^{2}-8x+16. 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}.

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

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

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

Whakarūnātia.

$$x-4=2 x-4=-2$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=6 x=2$$