Whakaoti i te (2x+3)(x^2-16)+(x-4)(x+40)=2(x-4)(x^2-16)

$$( 2 x + 3 ) ( x ^ { 2 } - 16 ) + ( x - 4 ) ( x + 40 ) = 2 ( x - 4 ) ( x ^ { 2 } - 16 )$$

Whakaoti mō x

$$x=-7$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+3 ki te x^{2}-16.

$$2x^{3}-32x+3x^{2}-48+\left(x-4\right)\left(x+40\right)=2\left(x-4\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te x+40 ka whakakotahi i ngā kupu rite.

$$2x^{3}-32x+3x^{2}-48+x^{2}+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}-32x+4x^{2}-48+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

Pahekotia te -32x me 36x, ka 4x.

$$2x^{3}+4x+4x^{2}-48-160=2\left(x-4\right)\left(x^{2}-16\right)$$

Tangohia te 160 i te -48, ka -208.

$$2x^{3}+4x+4x^{2}-208=2\left(x-4\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-4.

$$2x^{3}+4x+4x^{2}-208=\left(2x-8\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-8 ki te x^{2}-16.

$$2x^{3}+4x+4x^{2}-208=2x^{3}-32x-8x^{2}+128$$

Tangohia te 2x^{3} mai i ngā taha e rua.

$$2x^{3}+4x+4x^{2}-208-2x^{3}=-32x-8x^{2}+128$$

Pahekotia te 2x^{3} me -2x^{3}, ka 0.

$$4x+4x^{2}-208=-32x-8x^{2}+128$$

Me tāpiri te 32x ki ngā taha e rua.

$$4x+4x^{2}-208+32x=-8x^{2}+128$$

Pahekotia te 4x me 32x, ka 36x.

$$36x+4x^{2}-208=-8x^{2}+128$$

Me tāpiri te 8x^{2} ki ngā taha e rua.

$$36x+4x^{2}-208+8x^{2}=128$$

Pahekotia te 4x^{2} me 8x^{2}, ka 12x^{2}.

$$36x+12x^{2}-208=128$$

Tangohia te 128 mai i ngā taha e rua.

$$36x+12x^{2}-208-128=0$$

Tangohia te 128 i te -208, ka -336.

$$36x+12x^{2}-336=0$$

Whakawehea ngā taha e rua ki te 12.

$$3x+x^{2}-28=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}+3x-28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=1\left(-28\right)=-28$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -28.

$$-1,28 -2,14 -4,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+28=27 -2+14=12 -4+7=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-4 b=7$$

Tuhia anō te x^{2}+3x-28 hei \left(x^{2}-4x\right)+\left(7x-28\right).

$$\left(x^{2}-4x\right)+\left(7x-28\right)$$

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(x-4\right)+7\left(x-4\right)$$

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

$$\left(x-4\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+7=0.

$$x=4 x=-7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+3 ki te x^{2}-16.

$$2x^{3}-32x+3x^{2}-48+\left(x-4\right)\left(x+40\right)=2\left(x-4\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te x+40 ka whakakotahi i ngā kupu rite.

$$2x^{3}-32x+3x^{2}-48+x^{2}+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}-32x+4x^{2}-48+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

Pahekotia te -32x me 36x, ka 4x.

$$2x^{3}+4x+4x^{2}-48-160=2\left(x-4\right)\left(x^{2}-16\right)$$

Tangohia te 160 i te -48, ka -208.

$$2x^{3}+4x+4x^{2}-208=2\left(x-4\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-4.

$$2x^{3}+4x+4x^{2}-208=\left(2x-8\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-8 ki te x^{2}-16.

$$2x^{3}+4x+4x^{2}-208=2x^{3}-32x-8x^{2}+128$$

Tangohia te 2x^{3} mai i ngā taha e rua.

$$2x^{3}+4x+4x^{2}-208-2x^{3}=-32x-8x^{2}+128$$

Pahekotia te 2x^{3} me -2x^{3}, ka 0.

$$4x+4x^{2}-208=-32x-8x^{2}+128$$

Me tāpiri te 32x ki ngā taha e rua.

$$4x+4x^{2}-208+32x=-8x^{2}+128$$

Pahekotia te 4x me 32x, ka 36x.

$$36x+4x^{2}-208=-8x^{2}+128$$

Me tāpiri te 8x^{2} ki ngā taha e rua.

$$36x+4x^{2}-208+8x^{2}=128$$

Pahekotia te 4x^{2} me 8x^{2}, ka 12x^{2}.

$$36x+12x^{2}-208=128$$

Tangohia te 128 mai i ngā taha e rua.

$$36x+12x^{2}-208-128=0$$

Tangohia te 128 i te -208, ka -336.

$$36x+12x^{2}-336=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$12x^{2}+36x-336=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 12 mō a, 36 mō b, me -336 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\times 12\left(-336\right)}}{2\times 12}$$

Pūrua 36.

$$x=\frac{-36±\sqrt{1296-4\times 12\left(-336\right)}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-36±\sqrt{1296-48\left(-336\right)}}{2\times 12}$$

Whakareatia -48 ki te -336.

$$x=\frac{-36±\sqrt{1296+16128}}{2\times 12}$$

Tāpiri 1296 ki te 16128.

$$x=\frac{-36±\sqrt{17424}}{2\times 12}$$

Tuhia te pūtakerua o te 17424.

$$x=\frac{-36±132}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{-36±132}{24}$$

Nā, me whakaoti te whārite x=\frac{-36±132}{24} ina he tāpiri te ±. Tāpiri -36 ki te 132.

$$x=\frac{96}{24}$$

Whakawehe 96 ki te 24.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-36±132}{24} ina he tango te ±. Tango 132 mai i -36.

$$x=-\frac{168}{24}$$

Whakawehe -168 ki te 24.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=4 x=-7$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+3 ki te x^{2}-16.

$$2x^{3}-32x+3x^{2}-48+\left(x-4\right)\left(x+40\right)=2\left(x-4\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-4 ki te x+40 ka whakakotahi i ngā kupu rite.

$$2x^{3}-32x+3x^{2}-48+x^{2}+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}-32x+4x^{2}-48+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

Pahekotia te -32x me 36x, ka 4x.

$$2x^{3}+4x+4x^{2}-48-160=2\left(x-4\right)\left(x^{2}-16\right)$$

Tangohia te 160 i te -48, ka -208.

$$2x^{3}+4x+4x^{2}-208=2\left(x-4\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-4.

$$2x^{3}+4x+4x^{2}-208=\left(2x-8\right)\left(x^{2}-16\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-8 ki te x^{2}-16.

$$2x^{3}+4x+4x^{2}-208=2x^{3}-32x-8x^{2}+128$$

Tangohia te 2x^{3} mai i ngā taha e rua.

$$2x^{3}+4x+4x^{2}-208-2x^{3}=-32x-8x^{2}+128$$

Pahekotia te 2x^{3} me -2x^{3}, ka 0.

$$4x+4x^{2}-208=-32x-8x^{2}+128$$

Me tāpiri te 32x ki ngā taha e rua.

$$4x+4x^{2}-208+32x=-8x^{2}+128$$

Pahekotia te 4x me 32x, ka 36x.

$$36x+4x^{2}-208=-8x^{2}+128$$

Me tāpiri te 8x^{2} ki ngā taha e rua.

$$36x+4x^{2}-208+8x^{2}=128$$

Pahekotia te 4x^{2} me 8x^{2}, ka 12x^{2}.

$$36x+12x^{2}-208=128$$

Me tāpiri te 208 ki ngā taha e rua.

$$36x+12x^{2}=128+208$$

Tāpirihia te 128 ki te 208, ka 336.

$$36x+12x^{2}=336$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$12x^{2}+36x=336$$

Whakawehea ngā taha e rua ki te 12.

$$\frac{12x^{2}+36x}{12}=\frac{336}{12}$$

Mā te whakawehe ki te 12 ka wetekia te whakareanga ki te 12.

$$x^{2}+\frac{36}{12}x=\frac{336}{12}$$

Whakawehe 36 ki te 12.

$$x^{2}+3x=\frac{336}{12}$$

Whakawehe 336 ki te 12.

$$x^{2}+3x=28$$

Whakawehea te 3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{2}. Nā, tāpiria te pūrua o te \frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=28+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=28+\frac{9}{4}$$

Tāpiri 28 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{121}{4}$$

Tauwehea x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{11}{2} x+\frac{3}{2}=-\frac{11}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=4 x=-7$$