Whakaoti i te 5L^2-14L+8

$$5 L ^ { 2 } - 14 L + 8$$

Tauwehe

$$\left(L-2\right)\left(5L-4\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 5L^{2}+aL+bL+8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=5\times 8=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-10 b=-4$$

Tuhia anō te 5L^{2}-14L+8 hei \left(5L^{2}-10L\right)+\left(-4L+8\right).

$$\left(5L^{2}-10L\right)+\left(-4L+8\right)$$

Tauwehea te 5L i te tuatahi me te -4 i te rōpū tuarua.

$$5L\left(L-2\right)-4\left(L-2\right)$$

Whakatauwehea atu te kīanga pātahi L-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(L-2\right)\left(5L-4\right)$$

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.

$$5L^{2}-14L+8=0$$

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.

$$L=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\times 8}}{2\times 5}$$

Pūrua -14.

$$L=\frac{-\left(-14\right)±\sqrt{196-4\times 5\times 8}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$L=\frac{-\left(-14\right)±\sqrt{196-20\times 8}}{2\times 5}$$

Whakareatia -20 ki te 8.

$$L=\frac{-\left(-14\right)±\sqrt{196-160}}{2\times 5}$$

Tāpiri 196 ki te -160.

$$L=\frac{-\left(-14\right)±\sqrt{36}}{2\times 5}$$

Tuhia te pūtakerua o te 36.

$$L=\frac{-\left(-14\right)±6}{2\times 5}$$

Ko te tauaro o -14 ko 14.

$$L=\frac{14±6}{2\times 5}$$

Whakareatia 2 ki te 5.

$$L=\frac{14±6}{10}$$

Nā, me whakaoti te whārite L=\frac{14±6}{10} ina he tāpiri te ±. Tāpiri 14 ki te 6.

$$L=\frac{20}{10}$$

Whakawehe 20 ki te 10.

$$L=2$$

Nā, me whakaoti te whārite L=\frac{14±6}{10} ina he tango te ±. Tango 6 mai i 14.

$$L=\frac{8}{10}$$

Whakahekea te hautanga \frac{8}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$L=\frac{4}{5}$$

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 2 mō te x_{1} me te \frac{4}{5} mō te x_{2}.

$$5L^{2}-14L+8=5\left(L-2\right)\left(L-\frac{4}{5}\right)$$

Tango \frac{4}{5} mai i L 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.

$$5L^{2}-14L+8=5\left(L-2\right)\times \frac{5L-4}{5}$$

Whakakorea atu te tauwehe pūnoa nui rawa 5 i roto i te 5 me te 5.

$$5L^{2}-14L+8=\left(L-2\right)\left(5L-4\right)$$

Aromātai

$$\left(L-2\right)\left(5L-4\right)$$