Whakaoti i te 7x^2-37x+10

$$7 { x }^{ 2 } -37x+10$$

Tauwehe

$$\left(x-5\right)\left(7x-2\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 7x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-37 ab=7\times 10=70$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 70.

$$-1,-70 -2,-35 -5,-14 -7,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-70=-71 -2-35=-37 -5-14=-19 -7-10=-17$$

Ko te otinga te takirua ka hoatu i te tapeke -37.

$$a=-35 b=-2$$

Tuhia anō te 7x^{2}-37x+10 hei \left(7x^{2}-35x\right)+\left(-2x+10\right).

$$\left(7x^{2}-35x\right)+\left(-2x+10\right)$$

Tauwehea te 7x i te tuatahi me te -2 i te rōpū tuarua.

$$7x\left(x-5\right)-2\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(7x-2\right)$$

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.

$$7x^{2}-37x+10=0$$

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.

$$x=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 7\times 10}}{2\times 7}$$

Pūrua -37.

$$x=\frac{-\left(-37\right)±\sqrt{1369-4\times 7\times 10}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-37\right)±\sqrt{1369-28\times 10}}{2\times 7}$$

Whakareatia -28 ki te 10.

$$x=\frac{-\left(-37\right)±\sqrt{1369-280}}{2\times 7}$$

Tāpiri 1369 ki te -280.

$$x=\frac{-\left(-37\right)±\sqrt{1089}}{2\times 7}$$

Tuhia te pūtakerua o te 1089.

$$x=\frac{-\left(-37\right)±33}{2\times 7}$$

Ko te tauaro o -37 ko 37.

$$x=\frac{37±33}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{37±33}{14}$$

Nā, me whakaoti te whārite x=\frac{37±33}{14} ina he tāpiri te ±. Tāpiri 37 ki te 33.

$$x=\frac{70}{14}$$

Whakawehe 70 ki te 14.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{37±33}{14} ina he tango te ±. Tango 33 mai i 37.

$$x=\frac{4}{14}$$

Whakahekea te hautanga \frac{4}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{2}{7}$$

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 5 mō te x_{1} me te \frac{2}{7} mō te x_{2}.

$$7x^{2}-37x+10=7\left(x-5\right)\left(x-\frac{2}{7}\right)$$

Tango \frac{2}{7} 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.

$$7x^{2}-37x+10=7\left(x-5\right)\times \frac{7x-2}{7}$$

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

$$7x^{2}-37x+10=\left(x-5\right)\left(7x-2\right)$$

Aromātai

$$\left(x-5\right)\left(7x-2\right)$$