Whakaoti i te x^2-x+12=2x^2+3x+7

$$x ^ { 2 } - x + 12 = 2 x ^ { 2 } + 3 x + 7$$

Whakaoti mō x

$$x=-5$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$x^{2}-x+12-2x^{2}=3x+7$$

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

$$-x^{2}-x+12=3x+7$$

Tangohia te 3x mai i ngā taha e rua.

$$-x^{2}-x+12-3x=7$$

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

$$-x^{2}-4x+12=7$$

Tangohia te 7 mai i ngā taha e rua.

$$-x^{2}-4x+12-7=0$$

Tangohia te 7 i te 12, ka 5.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

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

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=-5$$

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

$$\left(-x^{2}+x\right)+\left(-5x+5\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(-x+1\right)+5\left(-x+1\right)$$

Whakatauwehea atu te kīanga pātahi -x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+1\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te -x+1=0 me te x+5=0.

$$x=1 x=-5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$x^{2}-x+12-2x^{2}=3x+7$$

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

$$-x^{2}-x+12=3x+7$$

Tangohia te 3x mai i ngā taha e rua.

$$-x^{2}-x+12-3x=7$$

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

$$-x^{2}-4x+12=7$$

Tangohia te 7 mai i ngā taha e rua.

$$-x^{2}-4x+12-7=0$$

Tangohia te 7 i te 12, ka 5.

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

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

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 5}}{2\left(-1\right)}$$

Whakareatia 4 ki te 5.

$$x=\frac{-\left(-4\right)±\sqrt{16+20}}{2\left(-1\right)}$$

Tāpiri 16 ki te 20.

$$x=\frac{-\left(-4\right)±\sqrt{36}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-4\right)±6}{2\left(-1\right)}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±6}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{4±6}{-2}$$

Nā, me whakaoti te whārite x=\frac{4±6}{-2} ina he tāpiri te ±. Tāpiri 4 ki te 6.

$$x=\frac{10}{-2}$$

Whakawehe 10 ki te -2.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{4±6}{-2} ina he tango te ±. Tango 6 mai i 4.

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

Whakawehe -2 ki te -2.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=-5 x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$x^{2}-x+12-2x^{2}=3x+7$$

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

$$-x^{2}-x+12=3x+7$$

Tangohia te 3x mai i ngā taha e rua.

$$-x^{2}-x+12-3x=7$$

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

$$-x^{2}-4x+12=7$$

Tangohia te 12 mai i ngā taha e rua.

$$-x^{2}-4x=7-12$$

Tangohia te 12 i te 7, ka -5.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{5}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{5}{-1}$$

Whakawehe -4 ki te -1.

$$x^{2}+4x=-\frac{5}{-1}$$

Whakawehe -5 ki te -1.

$$x^{2}+4x=5$$

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

$$x^{2}+4x+2^{2}=5+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=5+4$$

Tāpiri 5 ki te 4.

$$x^{2}+4x+4=9$$

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}.

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

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x+2=3 x+2=-3$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=1 x=-5$$