Whakaoti i te x^2+11x+24

$$x^2+11x+24$$

Tauwehe

$$\left(x+3\right)\left(x+8\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=1\times 24=24$$

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 24.

$$1,24 2,12 3,8 4,6$$

Tātaihia te tapeke mō ia takirua.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=3 b=8$$

Tuhia anō te x^{2}+11x+24 hei \left(x^{2}+3x\right)+\left(8x+24\right).

$$\left(x^{2}+3x\right)+\left(8x+24\right)$$

Tauwehea te x i te tuatahi me te 8 i te rōpū tuarua.

$$x\left(x+3\right)+8\left(x+3\right)$$

Whakatauwehea atu te kīanga pātahi x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+3\right)\left(x+8\right)$$

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.

$$x^{2}+11x+24=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 24}}{2}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\times 24}}{2}$$

Whakareatia -4 ki te 24.

$$x=\frac{-11±\sqrt{121-96}}{2}$$

Tāpiri 121 ki te -96.

$$x=\frac{-11±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-11±5}{2}$$

Nā, me whakaoti te whārite x=\frac{-11±5}{2} ina he tāpiri te ±. Tāpiri -11 ki te 5.

$$x=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-11±5}{2} ina he tango te ±. Tango 5 mai i -11.

$$x=-\frac{16}{2}$$

Whakawehe -16 ki te 2.

$$x=-8$$

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 -3 mō te x_{1} me te -8 mō te x_{2}.

$$x^{2}+11x+24=\left(x-\left(-3\right)\right)\left(x-\left(-8\right)\right)$$

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

$$x^{2}+11x+24=\left(x+3\right)\left(x+8\right)$$

Aromātai

$$\left(x+3\right)\left(x+8\right)$$