Whakaoti i te 3x(x+1)-x=33-(x-3)^2

$$3 x ( x + 1 ) - x = 33 - ( x - 3 ) ^ { 2 }$$

Whakaoti mō x

$$x=-2$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$3x^{2}+3x-x=33-\left(x-3\right)^{2}$$

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

$$3x^{2}+2x=33-\left(x-3\right)^{2}$$

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

$$3x^{2}+2x=33-\left(x^{2}-6x+9\right)$$

Hei kimi i te tauaro o x^{2}-6x+9, kimihia te tauaro o ia taurangi.

$$3x^{2}+2x=33-x^{2}+6x-9$$

Tangohia te 9 i te 33, ka 24.

$$3x^{2}+2x=24-x^{2}+6x$$

Tangohia te 24 mai i ngā taha e rua.

$$3x^{2}+2x-24=-x^{2}+6x$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$3x^{2}+2x-24+x^{2}=6x$$

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

$$4x^{2}+2x-24=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$4x^{2}+2x-24-6x=0$$

Pahekotia te 2x me -6x, ka -4x.

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

Whakawehea ngā taha e rua ki te 4.

$$x^{2}-x-6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-6\right)=-6$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -6.

$$1,-6 2,-3$$

Tātaihia te tapeke mō ia takirua.

$$1-6=-5 2-3=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-3 b=2$$

Tuhia anō te x^{2}-x-6 hei \left(x^{2}-3x\right)+\left(2x-6\right).

$$\left(x^{2}-3x\right)+\left(2x-6\right)$$

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

$$x\left(x-3\right)+2\left(x-3\right)$$

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-3\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te x+2=0.

$$x=3 x=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$3x^{2}+3x-x=33-\left(x-3\right)^{2}$$

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

$$3x^{2}+2x=33-\left(x-3\right)^{2}$$

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

$$3x^{2}+2x=33-\left(x^{2}-6x+9\right)$$

Hei kimi i te tauaro o x^{2}-6x+9, kimihia te tauaro o ia taurangi.

$$3x^{2}+2x=33-x^{2}+6x-9$$

Tangohia te 9 i te 33, ka 24.

$$3x^{2}+2x=24-x^{2}+6x$$

Tangohia te 24 mai i ngā taha e rua.

$$3x^{2}+2x-24=-x^{2}+6x$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$3x^{2}+2x-24+x^{2}=6x$$

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

$$4x^{2}+2x-24=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$4x^{2}+2x-24-6x=0$$

Pahekotia te 2x me -6x, ka -4x.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -4 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\left(-24\right)}}{2\times 4}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-24\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\left(-24\right)}}{2\times 4}$$

Whakareatia -16 ki te -24.

$$x=\frac{-\left(-4\right)±\sqrt{16+384}}{2\times 4}$$

Tāpiri 16 ki te 384.

$$x=\frac{-\left(-4\right)±\sqrt{400}}{2\times 4}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-4\right)±20}{2\times 4}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±20}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{4±20}{8}$$

Nā, me whakaoti te whārite x=\frac{4±20}{8} ina he tāpiri te ±. Tāpiri 4 ki te 20.

$$x=\frac{24}{8}$$

Whakawehe 24 ki te 8.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{4±20}{8} ina he tango te ±. Tango 20 mai i 4.

$$x=-\frac{16}{8}$$

Whakawehe -16 ki te 8.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=3 x=-2$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$3x^{2}+3x-x=33-\left(x-3\right)^{2}$$

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

$$3x^{2}+2x=33-\left(x-3\right)^{2}$$

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

$$3x^{2}+2x=33-\left(x^{2}-6x+9\right)$$

Hei kimi i te tauaro o x^{2}-6x+9, kimihia te tauaro o ia taurangi.

$$3x^{2}+2x=33-x^{2}+6x-9$$

Tangohia te 9 i te 33, ka 24.

$$3x^{2}+2x=24-x^{2}+6x$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$3x^{2}+2x+x^{2}=24+6x$$

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

$$4x^{2}+2x=24+6x$$

Tangohia te 6x mai i ngā taha e rua.

$$4x^{2}+2x-6x=24$$

Pahekotia te 2x me -6x, ka -4x.

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

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-4x}{4}=\frac{24}{4}$$

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

$$x^{2}+\left(-\frac{4}{4}\right)x=\frac{24}{4}$$

Whakawehe -4 ki te 4.

$$x^{2}-x=\frac{24}{4}$$

Whakawehe 24 ki te 4.

$$x^{2}-x=6$$

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=6+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=6+\frac{1}{4}$$

Tāpiri 6 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{25}{4}$$

Tauwehea x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=3 x=-2$$