Whakaoti i te 196=2x^2+16+8x+x^2+4x

$$196 = 2 x ^ { 2 } + 16 + 8 x + x ^ { 2 } + 4 x$$

Whakaoti mō x

$$x=-10$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$196=3x^{2}+16+8x+4x$$

Pahekotia te 8x me 4x, ka 12x.

$$196=3x^{2}+16+12x$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x^{2}+16+12x=196$$

Tangohia te 196 mai i ngā taha e rua.

$$3x^{2}+16+12x-196=0$$

Tangohia te 196 i te 16, ka -180.

$$3x^{2}-180+12x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x^{2}-60+4x=0$$

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

$$x^{2}+4x-60=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-60. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=1\left(-60\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-6 b=10$$

Tuhia anō te x^{2}+4x-60 hei \left(x^{2}-6x\right)+\left(10x-60\right).

$$\left(x^{2}-6x\right)+\left(10x-60\right)$$

Tauwehea te x i te tuatahi me te 10 i te rōpū tuarua.

$$x\left(x-6\right)+10\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x+10=0.

$$x=6 x=-10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$196=3x^{2}+16+8x+4x$$

Pahekotia te 8x me 4x, ka 12x.

$$196=3x^{2}+16+12x$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x^{2}+16+12x=196$$

Tangohia te 196 mai i ngā taha e rua.

$$3x^{2}+16+12x-196=0$$

Tangohia te 196 i te 16, ka -180.

$$3x^{2}-180+12x=0$$

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.

$$3x^{2}+12x-180=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 12 mō b, me -180 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 3\left(-180\right)}}{2\times 3}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\times 3\left(-180\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-12±\sqrt{144-12\left(-180\right)}}{2\times 3}$$

Whakareatia -12 ki te -180.

$$x=\frac{-12±\sqrt{144+2160}}{2\times 3}$$

Tāpiri 144 ki te 2160.

$$x=\frac{-12±\sqrt{2304}}{2\times 3}$$

Tuhia te pūtakerua o te 2304.

$$x=\frac{-12±48}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-12±48}{6}$$

Nā, me whakaoti te whārite x=\frac{-12±48}{6} ina he tāpiri te ±. Tāpiri -12 ki te 48.

$$x=\frac{36}{6}$$

Whakawehe 36 ki te 6.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{-12±48}{6} ina he tango te ±. Tango 48 mai i -12.

$$x=-\frac{60}{6}$$

Whakawehe -60 ki te 6.

$$x=-10$$

Kua oti te whārite te whakatau.

$$x=6 x=-10$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$196=3x^{2}+16+8x+4x$$

Pahekotia te 8x me 4x, ka 12x.

$$196=3x^{2}+16+12x$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x^{2}+16+12x=196$$

Tangohia te 16 mai i ngā taha e rua.

$$3x^{2}+12x=196-16$$

Tangohia te 16 i te 196, ka 180.

$$3x^{2}+12x=180$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+12x}{3}=\frac{180}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{12}{3}x=\frac{180}{3}$$

Whakawehe 12 ki te 3.

$$x^{2}+4x=\frac{180}{3}$$

Whakawehe 180 ki te 3.

$$x^{2}+4x=60$$

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.

$$x^{2}+4x+2^{2}=60+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=60+4$$

Tāpiri 60 ki te 4.

$$x^{2}+4x+4=64$$

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}.

$$\left(x+2\right)^{2}=64$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$x+2=8 x+2=-8$$

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

$$x=6 x=-10$$