Whakaoti i te 7x^2-74x-120

$$7 { x }^{ 2 } -74x-120$$

Tauwehe

$$\left(x-12\right)\left(7x+10\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 7x^{2}+ax+bx-120. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-74 ab=7\left(-120\right)=-840$$

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 -840.

$$1,-840 2,-420 3,-280 4,-210 5,-168 6,-140 7,-120 8,-105 10,-84 12,-70 14,-60 15,-56 20,-42 21,-40 24,-35 28,-30$$

Tātaihia te tapeke mō ia takirua.

$$1-840=-839 2-420=-418 3-280=-277 4-210=-206 5-168=-163 6-140=-134 7-120=-113 8-105=-97 10-84=-74 12-70=-58 14-60=-46 15-56=-41 20-42=-22 21-40=-19 24-35=-11 28-30=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -74.

$$a=-84 b=10$$

Tuhia anō te 7x^{2}-74x-120 hei \left(7x^{2}-84x\right)+\left(10x-120\right).

$$\left(7x^{2}-84x\right)+\left(10x-120\right)$$

Tauwehea te 7x i te tuatahi me te 10 i te rōpū tuarua.

$$7x\left(x-12\right)+10\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(7x+10\right)$$

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.

$$7x^{2}-74x-120=0$$

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.

$$x=\frac{-\left(-74\right)±\sqrt{\left(-74\right)^{2}-4\times 7\left(-120\right)}}{2\times 7}$$

Pūrua -74.

$$x=\frac{-\left(-74\right)±\sqrt{5476-4\times 7\left(-120\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-74\right)±\sqrt{5476-28\left(-120\right)}}{2\times 7}$$

Whakareatia -28 ki te -120.

$$x=\frac{-\left(-74\right)±\sqrt{5476+3360}}{2\times 7}$$

Tāpiri 5476 ki te 3360.

$$x=\frac{-\left(-74\right)±\sqrt{8836}}{2\times 7}$$

Tuhia te pūtakerua o te 8836.

$$x=\frac{-\left(-74\right)±94}{2\times 7}$$

Ko te tauaro o -74 ko 74.

$$x=\frac{74±94}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{74±94}{14}$$

Nā, me whakaoti te whārite x=\frac{74±94}{14} ina he tāpiri te ±. Tāpiri 74 ki te 94.

$$x=\frac{168}{14}$$

Whakawehe 168 ki te 14.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{74±94}{14} ina he tango te ±. Tango 94 mai i 74.

$$x=-\frac{20}{14}$$

Whakahekea te hautanga \frac{-20}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{10}{7}$$

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 12 mō te x_{1} me te -\frac{10}{7} mō te x_{2}.

$$7x^{2}-74x-120=7\left(x-12\right)\left(x-\left(-\frac{10}{7}\right)\right)$$

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

$$7x^{2}-74x-120=7\left(x-12\right)\left(x+\frac{10}{7}\right)$$

Tāpiri \frac{10}{7} ki te x 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.

$$7x^{2}-74x-120=7\left(x-12\right)\times \frac{7x+10}{7}$$

Whakakorea atu te tauwehe pūnoa nui rawa 7 i roto i te 7 me te 7.

$$7x^{2}-74x-120=\left(x-12\right)\left(7x+10\right)$$

Aromātai

$$\left(x-12\right)\left(7x+10\right)$$