Whakaoti i te 2x/x-4+3/x-3+4=30+5x^2-36x/x^2-7x+12

$$\frac { 2 x } { x - 4 } + \frac { 3 } { x - 3 } + 4 = \frac { 30 + 5 x ^ { 2 } - 36 x } { x ^ { 2 } - 7 x + 12 }$$

Whakaoti mō x

$$x=-3$$

$$x=-2$$

Ngā Upane Whakamahi Tauwehe

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-3 ki te 2.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-6 ki te x.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-4 ki te 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Pahekotia te -6x me 3x, ka -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-7x+12 ki te 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Pahekotia te 2x^{2} me 4x^{2}, ka 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Pahekotia te -3x me -28x, ka -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Tāpirihia te -12 ki te 48, ka 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Tangohia te 30 mai i ngā taha e rua.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

Tangohia te 30 i te 36, ka 6.

$$6x^{2}-31x+6=5x^{2}-36x$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$6x^{2}-31x+6-5x^{2}=-36x$$

Pahekotia te 6x^{2} me -5x^{2}, ka x^{2}.

$$x^{2}-31x+6=-36x$$

Me tāpiri te 36x ki ngā taha e rua.

$$x^{2}-31x+6+36x=0$$

Pahekotia te -31x me 36x, ka 5x.

$$x^{2}+5x+6=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+5x+6 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.

$$a+b=5 ab=6$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 6.

$$1,6 2,3$$

Tātaihia te tapeke mō ia takirua.

$$1+6=7 2+3=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=2 b=3$$

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

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

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

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-3 ki te 2.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-6 ki te x.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-4 ki te 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Pahekotia te -6x me 3x, ka -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-7x+12 ki te 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Pahekotia te 2x^{2} me 4x^{2}, ka 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Pahekotia te -3x me -28x, ka -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Tāpirihia te -12 ki te 48, ka 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Tangohia te 30 mai i ngā taha e rua.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

Tangohia te 30 i te 36, ka 6.

$$6x^{2}-31x+6=5x^{2}-36x$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$6x^{2}-31x+6-5x^{2}=-36x$$

Pahekotia te 6x^{2} me -5x^{2}, ka x^{2}.

$$x^{2}-31x+6=-36x$$

Me tāpiri te 36x ki ngā taha e rua.

$$x^{2}-31x+6+36x=0$$

Pahekotia te -31x me 36x, ka 5x.

$$x^{2}+5x+6=0$$

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.

$$a+b=5 ab=1\times 6=6$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 6.

$$1,6 2,3$$

Tātaihia te tapeke mō ia takirua.

$$1+6=7 2+3=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=2 b=3$$

Tuhia anō te x^{2}+5x+6 hei \left(x^{2}+2x\right)+\left(3x+6\right).

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

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

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

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

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

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-3 ki te 2.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-6 ki te x.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-4 ki te 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Pahekotia te -6x me 3x, ka -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-7x+12 ki te 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Pahekotia te 2x^{2} me 4x^{2}, ka 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Pahekotia te -3x me -28x, ka -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Tāpirihia te -12 ki te 48, ka 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Tangohia te 30 mai i ngā taha e rua.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

Tangohia te 30 i te 36, ka 6.

$$6x^{2}-31x+6=5x^{2}-36x$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$6x^{2}-31x+6-5x^{2}=-36x$$

Pahekotia te 6x^{2} me -5x^{2}, ka x^{2}.

$$x^{2}-31x+6=-36x$$

Me tāpiri te 36x ki ngā taha e rua.

$$x^{2}-31x+6+36x=0$$

Pahekotia te -31x me 36x, ka 5x.

$$x^{2}+5x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 5 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 6}}{2}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 6}}{2}$$

Whakareatia -4 ki te 6.

$$x=\frac{-5±\sqrt{25-24}}{2}$$

Tāpiri 25 ki te -24.

$$x=\frac{-5±\sqrt{1}}{2}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-5±1}{2}$$

Nā, me whakaoti te whārite x=\frac{-5±1}{2} ina he tāpiri te ±. Tāpiri -5 ki te 1.

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

Whakawehe -4 ki te 2.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-5±1}{2} ina he tango te ±. Tango 1 mai i -5.

$$x=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$x=-3$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-3 ki te 2.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-6 ki te x.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-4 ki te 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Pahekotia te -6x me 3x, ka -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-7x+12 ki te 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Pahekotia te 2x^{2} me 4x^{2}, ka 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Pahekotia te -3x me -28x, ka -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Tāpirihia te -12 ki te 48, ka 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$6x^{2}-31x+36-5x^{2}=30-36x$$

Pahekotia te 6x^{2} me -5x^{2}, ka x^{2}.

$$x^{2}-31x+36=30-36x$$

Me tāpiri te 36x ki ngā taha e rua.

$$x^{2}-31x+36+36x=30$$

Pahekotia te -31x me 36x, ka 5x.

$$x^{2}+5x+36=30$$

Tangohia te 36 mai i ngā taha e rua.

$$x^{2}+5x=30-36$$

Tangohia te 36 i te 30, ka -6.

$$x^{2}+5x=-6$$

Whakawehea te 5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{2}. Nā, tāpiria te pūrua o te \frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-6+\left(\frac{5}{2}\right)^{2}$$

Pūruatia \frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+5x+\frac{25}{4}=-6+\frac{25}{4}$$

Tāpiri -6 ki te \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{1}{4}$$

Tauwehea x^{2}+5x+\frac{25}{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}.

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

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

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

Whakarūnātia.

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

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

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