Whakaoti i te 6r^2+29r-42

$$6 r ^ { 2 } + 29 r - 42$$

Tauwehe

$$\left(6r-7\right)\left(r+6\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6r^{2}+ar+br-42. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=29 ab=6\left(-42\right)=-252$$

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 -252.

$$-1,252 -2,126 -3,84 -4,63 -6,42 -7,36 -9,28 -12,21 -14,18$$

Tātaihia te tapeke mō ia takirua.

$$-1+252=251 -2+126=124 -3+84=81 -4+63=59 -6+42=36 -7+36=29 -9+28=19 -12+21=9 -14+18=4$$

Ko te otinga te takirua ka hoatu i te tapeke 29.

$$a=-7 b=36$$

Tuhia anō te 6r^{2}+29r-42 hei \left(6r^{2}-7r\right)+\left(36r-42\right).

$$\left(6r^{2}-7r\right)+\left(36r-42\right)$$

Tauwehea te r i te tuatahi me te 6 i te rōpū tuarua.

$$r\left(6r-7\right)+6\left(6r-7\right)$$

Whakatauwehea atu te kīanga pātahi 6r-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6r-7\right)\left(r+6\right)$$

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.

$$6r^{2}+29r-42=0$$

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.

$$r=\frac{-29±\sqrt{29^{2}-4\times 6\left(-42\right)}}{2\times 6}$$

Pūrua 29.

$$r=\frac{-29±\sqrt{841-4\times 6\left(-42\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$r=\frac{-29±\sqrt{841-24\left(-42\right)}}{2\times 6}$$

Whakareatia -24 ki te -42.

$$r=\frac{-29±\sqrt{841+1008}}{2\times 6}$$

Tāpiri 841 ki te 1008.

$$r=\frac{-29±\sqrt{1849}}{2\times 6}$$

Tuhia te pūtakerua o te 1849.

$$r=\frac{-29±43}{2\times 6}$$

Whakareatia 2 ki te 6.

$$r=\frac{-29±43}{12}$$

Nā, me whakaoti te whārite r=\frac{-29±43}{12} ina he tāpiri te ±. Tāpiri -29 ki te 43.

$$r=\frac{14}{12}$$

Whakahekea te hautanga \frac{14}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$r=\frac{7}{6}$$

Nā, me whakaoti te whārite r=\frac{-29±43}{12} ina he tango te ±. Tango 43 mai i -29.

$$r=-\frac{72}{12}$$

Whakawehe -72 ki te 12.

$$r=-6$$

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{7}{6} mō te x_{1} me te -6 mō te x_{2}.

$$6r^{2}+29r-42=6\left(r-\frac{7}{6}\right)\left(r-\left(-6\right)\right)$$

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

$$6r^{2}+29r-42=6\left(r-\frac{7}{6}\right)\left(r+6\right)$$

Tango \frac{7}{6} mai i r 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.

$$6r^{2}+29r-42=6\times \frac{6r-7}{6}\left(r+6\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 6 i roto i te 6 me te 6.

$$6r^{2}+29r-42=\left(6r-7\right)\left(r+6\right)$$

Aromātai

$$\left(6r-7\right)\left(r+6\right)$$