Whakaoti i te (x-4)(3x+6)+(x-4)(12x+48)=0

$$( x - 4 ) ( 3 x + 6 ) + ( x - 4 ) ( 12 x + 48 ) = 0$$

Whakaoti mō x

$$x = -\frac{18}{5} = -3\frac{3}{5} = -3.6$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te 3x+6 ka whakakotahi i ngā kupu rite.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te 12x+48 ka whakakotahi i ngā kupu rite.

$$3x^{2}-6x-24+12x^{2}-192=0$$

Pahekotia te 3x^{2} me 12x^{2}, ka 15x^{2}.

$$15x^{2}-6x-24-192=0$$

Tangohia te 192 i te -24, ka -216.

$$15x^{2}-6x-216=0$$

Whakawehea ngā taha e rua ki te 3.

$$5x^{2}-2x-72=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5x^{2}+ax+bx-72. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=5\left(-72\right)=-360$$

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 -360.

$$1,-360 2,-180 3,-120 4,-90 5,-72 6,-60 8,-45 9,-40 10,-36 12,-30 15,-24 18,-20$$

Tātaihia te tapeke mō ia takirua.

$$1-360=-359 2-180=-178 3-120=-117 4-90=-86 5-72=-67 6-60=-54 8-45=-37 9-40=-31 10-36=-26 12-30=-18 15-24=-9 18-20=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-20 b=18$$

Tuhia anō te 5x^{2}-2x-72 hei \left(5x^{2}-20x\right)+\left(18x-72\right).

$$\left(5x^{2}-20x\right)+\left(18x-72\right)$$

Tauwehea te 5x i te tuatahi me te 18 i te rōpū tuarua.

$$5x\left(x-4\right)+18\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(5x+18\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te 5x+18=0.

$$x=4 x=-\frac{18}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te 3x+6 ka whakakotahi i ngā kupu rite.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te 12x+48 ka whakakotahi i ngā kupu rite.

$$3x^{2}-6x-24+12x^{2}-192=0$$

Pahekotia te 3x^{2} me 12x^{2}, ka 15x^{2}.

$$15x^{2}-6x-24-192=0$$

Tangohia te 192 i te -24, ka -216.

$$15x^{2}-6x-216=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, -6 mō b, me -216 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 15\left(-216\right)}}{2\times 15}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 15\left(-216\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-\left(-6\right)±\sqrt{36-60\left(-216\right)}}{2\times 15}$$

Whakareatia -60 ki te -216.

$$x=\frac{-\left(-6\right)±\sqrt{36+12960}}{2\times 15}$$

Tāpiri 36 ki te 12960.

$$x=\frac{-\left(-6\right)±\sqrt{12996}}{2\times 15}$$

Tuhia te pūtakerua o te 12996.

$$x=\frac{-\left(-6\right)±114}{2\times 15}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±114}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{6±114}{30}$$

Nā, me whakaoti te whārite x=\frac{6±114}{30} ina he tāpiri te ±. Tāpiri 6 ki te 114.

$$x=\frac{120}{30}$$

Whakawehe 120 ki te 30.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{6±114}{30} ina he tango te ±. Tango 114 mai i 6.

$$x=-\frac{108}{30}$$

Whakahekea te hautanga \frac{-108}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{18}{5}$$

Kua oti te whārite te whakatau.

$$x=4 x=-\frac{18}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te 3x+6 ka whakakotahi i ngā kupu rite.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te 12x+48 ka whakakotahi i ngā kupu rite.

$$3x^{2}-6x-24+12x^{2}-192=0$$

Pahekotia te 3x^{2} me 12x^{2}, ka 15x^{2}.

$$15x^{2}-6x-24-192=0$$

Tangohia te 192 i te -24, ka -216.

$$15x^{2}-6x-216=0$$

Me tāpiri te 216 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$15x^{2}-6x=216$$

Whakawehea ngā taha e rua ki te 15.

$$\frac{15x^{2}-6x}{15}=\frac{216}{15}$$

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

$$x^{2}+\left(-\frac{6}{15}\right)x=\frac{216}{15}$$

Whakahekea te hautanga \frac{-6}{15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{2}{5}x=\frac{216}{15}$$

Whakahekea te hautanga \frac{216}{15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{2}{5}x=\frac{72}{5}$$

Whakawehea te -\frac{2}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{5}. Nā, tāpiria te pūrua o te -\frac{1}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=\frac{72}{5}+\left(-\frac{1}{5}\right)^{2}$$

Pūruatia -\frac{1}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{72}{5}+\frac{1}{25}$$

Tāpiri \frac{72}{5} ki te \frac{1}{25} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{361}{25}$$

Tauwehea x^{2}-\frac{2}{5}x+\frac{1}{25}. 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}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{361}{25}$$

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

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{361}{25}}$$

Whakarūnātia.

$$x-\frac{1}{5}=\frac{19}{5} x-\frac{1}{5}=-\frac{19}{5}$$

Me tāpiri \frac{1}{5} ki ngā taha e rua o te whārite.

$$x=4 x=-\frac{18}{5}$$