Whakaoti i te 9x^2+39x+42=

$$9 x ^ { 2 } + 39 x + 42 =$$

Tauwehe

$$3\left(x+2\right)\left(3x+7\right)$$

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(3x^{2}+13x+14\right)$$

Whakaarohia te 3x^{2}+13x+14. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 3x^{2}+ax+bx+14. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=3\times 14=42$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 42.

$$1,42 2,21 3,14 6,7$$

Tātaihia te tapeke mō ia takirua.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=6 b=7$$

Tuhia anō te 3x^{2}+13x+14 hei \left(3x^{2}+6x\right)+\left(7x+14\right).

$$\left(3x^{2}+6x\right)+\left(7x+14\right)$$

Tauwehea te 3x i te tuatahi me te 7 i te rōpū tuarua.

$$3x\left(x+2\right)+7\left(x+2\right)$$

Whakatauwehea atu te kīanga pātahi x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+2\right)\left(3x+7\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$3\left(x+2\right)\left(3x+7\right)$$

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.

$$9x^{2}+39x+42=0$$

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.

$$x=\frac{-39±\sqrt{39^{2}-4\times 9\times 42}}{2\times 9}$$

Pūrua 39.

$$x=\frac{-39±\sqrt{1521-4\times 9\times 42}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-39±\sqrt{1521-36\times 42}}{2\times 9}$$

Whakareatia -36 ki te 42.

$$x=\frac{-39±\sqrt{1521-1512}}{2\times 9}$$

Tāpiri 1521 ki te -1512.

$$x=\frac{-39±\sqrt{9}}{2\times 9}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-39±3}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-39±3}{18}$$

Nā, me whakaoti te whārite x=\frac{-39±3}{18} ina he tāpiri te ±. Tāpiri -39 ki te 3.

$$x=-\frac{36}{18}$$

Whakawehe -36 ki te 18.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-39±3}{18} ina he tango te ±. Tango 3 mai i -39.

$$x=-\frac{42}{18}$$

Whakahekea te hautanga \frac{-42}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{7}{3}$$

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{7}{3} mō te x_{2}.

$$9x^{2}+39x+42=9\left(x-\left(-2\right)\right)\left(x-\left(-\frac{7}{3}\right)\right)$$

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

$$9x^{2}+39x+42=9\left(x+2\right)\left(x+\frac{7}{3}\right)$$

Tāpiri \frac{7}{3} 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.

$$9x^{2}+39x+42=9\left(x+2\right)\times \frac{3x+7}{3}$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te 9 me te 3.

$$9x^{2}+39x+42=3\left(x+2\right)\left(3x+7\right)$$

Aromātai

$$3\left(x+2\right)\left(3x+7\right)$$