Whakaoti i te 4q^2-34q+70

$$4 q ^ { 2 } - 34 q + 70$$

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 2.

$$2\left(2q^{2}-17q+35\right)$$

Whakaarohia te 2q^{2}-17q+35. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2q^{2}+aq+bq+35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-17 ab=2\times 35=70$$

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 -17.

$$a=-10 b=-7$$

Tuhia anō te 2q^{2}-17q+35 hei \left(2q^{2}-10q\right)+\left(-7q+35\right).

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

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

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

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

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.

$$4q^{2}-34q+70=0$$

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.

$$q=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 4\times 70}}{2\times 4}$$

Pūrua -34.

$$q=\frac{-\left(-34\right)±\sqrt{1156-4\times 4\times 70}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$q=\frac{-\left(-34\right)±\sqrt{1156-16\times 70}}{2\times 4}$$

Whakareatia -16 ki te 70.

$$q=\frac{-\left(-34\right)±\sqrt{1156-1120}}{2\times 4}$$

Tāpiri 1156 ki te -1120.

$$q=\frac{-\left(-34\right)±\sqrt{36}}{2\times 4}$$

Tuhia te pūtakerua o te 36.

$$q=\frac{-\left(-34\right)±6}{2\times 4}$$

Ko te tauaro o -34 ko 34.

$$q=\frac{34±6}{2\times 4}$$

Whakareatia 2 ki te 4.

$$q=\frac{34±6}{8}$$

Nā, me whakaoti te whārite q=\frac{34±6}{8} ina he tāpiri te ±. Tāpiri 34 ki te 6.

$$q=\frac{40}{8}$$

Whakawehe 40 ki te 8.

$$q=5$$

Nā, me whakaoti te whārite q=\frac{34±6}{8} ina he tango te ±. Tango 6 mai i 34.

$$q=\frac{28}{8}$$

Whakahekea te hautanga \frac{28}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

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{7}{2} mō te x_{2}.

$$4q^{2}-34q+70=4\left(q-5\right)\left(q-\frac{7}{2}\right)$$

Tango \frac{7}{2} mai i q 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.

$$4q^{2}-34q+70=4\left(q-5\right)\times \frac{2q-7}{2}$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 4 me te 2.

$$4q^{2}-34q+70=2\left(q-5\right)\left(2q-7\right)$$

Aromātai

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