Whakaoti i te 11x^2+140x-196=

$$11 { x }^{ 2 } +140x-196=$$

Tauwehe

$$\left(11x-14\right)\left(x+14\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 11x^{2}+ax+bx-196. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=140 ab=11\left(-196\right)=-2156$$

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 -2156.

$$-1,2156 -2,1078 -4,539 -7,308 -11,196 -14,154 -22,98 -28,77 -44,49$$

Tātaihia te tapeke mō ia takirua.

$$-1+2156=2155 -2+1078=1076 -4+539=535 -7+308=301 -11+196=185 -14+154=140 -22+98=76 -28+77=49 -44+49=5$$

Ko te otinga te takirua ka hoatu i te tapeke 140.

$$a=-14 b=154$$

Tuhia anō te 11x^{2}+140x-196 hei \left(11x^{2}-14x\right)+\left(154x-196\right).

$$\left(11x^{2}-14x\right)+\left(154x-196\right)$$

Tauwehea te x i te tuatahi me te 14 i te rōpū tuarua.

$$x\left(11x-14\right)+14\left(11x-14\right)$$

Whakatauwehea atu te kīanga pātahi 11x-14 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(11x-14\right)\left(x+14\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$11x^{2}+140x-196=0$$

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.

$$x=\frac{-140±\sqrt{140^{2}-4\times 11\left(-196\right)}}{2\times 11}$$

Pūrua 140.

$$x=\frac{-140±\sqrt{19600-4\times 11\left(-196\right)}}{2\times 11}$$

Whakareatia -4 ki te 11.

$$x=\frac{-140±\sqrt{19600-44\left(-196\right)}}{2\times 11}$$

Whakareatia -44 ki te -196.

$$x=\frac{-140±\sqrt{19600+8624}}{2\times 11}$$

Tāpiri 19600 ki te 8624.

$$x=\frac{-140±\sqrt{28224}}{2\times 11}$$

Tuhia te pūtakerua o te 28224.

$$x=\frac{-140±168}{2\times 11}$$

Whakareatia 2 ki te 11.

$$x=\frac{-140±168}{22}$$

Nā, me whakaoti te whārite x=\frac{-140±168}{22} ina he tāpiri te ±. Tāpiri -140 ki te 168.

$$x=\frac{28}{22}$$

Whakahekea te hautanga \frac{28}{22} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{14}{11}$$

Nā, me whakaoti te whārite x=\frac{-140±168}{22} ina he tango te ±. Tango 168 mai i -140.

$$x=-\frac{308}{22}$$

Whakawehe -308 ki te 22.

$$x=-14$$

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te \frac{14}{11} mō te x_{1} me te -14 mō te x_{2}.

$$11x^{2}+140x-196=11\left(x-\frac{14}{11}\right)\left(x-\left(-14\right)\right)$$

Whakamāmātia ngā kīanga katoa o te āhua p-\left(-q\right) ki te p+q.

$$11x^{2}+140x-196=11\left(x-\frac{14}{11}\right)\left(x+14\right)$$

Tango \frac{14}{11} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$11x^{2}+140x-196=11\times \frac{11x-14}{11}\left(x+14\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 11 i roto i te 11 me te 11.

$$11x^{2}+140x-196=\left(11x-14\right)\left(x+14\right)$$

Aromātai

$$\left(11x-14\right)\left(x+14\right)$$