Whakaoti i te 14x+x^2+45

$$14 x + x ^ { 2 } + 45$$

Tauwehe

$$\left(x+5\right)\left(x+9\right)$$

Ngā Upane Otinga

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}+14x+45$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=1\times 45=45$$

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 45.

$$1,45 3,15 5,9$$

Tātaihia te tapeke mō ia takirua.

$$1+45=46 3+15=18 5+9=14$$

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

$$a=5 b=9$$

Tuhia anō te x^{2}+14x+45 hei \left(x^{2}+5x\right)+\left(9x+45\right).

$$\left(x^{2}+5x\right)+\left(9x+45\right)$$

Tauwehea te x i te tuatahi me te 9 i te rōpū tuarua.

$$x\left(x+5\right)+9\left(x+5\right)$$

Whakatauwehea atu te kīanga pātahi x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+5\right)\left(x+9\right)$$

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.

$$x^{2}+14x+45=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\times 45}}{2}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 45}}{2}$$

Whakareatia -4 ki te 45.

$$x=\frac{-14±\sqrt{196-180}}{2}$$

Tāpiri 196 ki te -180.

$$x=\frac{-14±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-14±4}{2}$$

Nā, me whakaoti te whārite x=\frac{-14±4}{2} ina he tāpiri te ±. Tāpiri -14 ki te 4.

$$x=-\frac{10}{2}$$

Whakawehe -10 ki te 2.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-14±4}{2} ina he tango te ±. Tango 4 mai i -14.

$$x=-\frac{18}{2}$$

Whakawehe -18 ki te 2.

$$x=-9$$

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 -9 mō te x_{2}.

$$x^{2}+14x+45=\left(x-\left(-5\right)\right)\left(x-\left(-9\right)\right)$$

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

$$x^{2}+14x+45=\left(x+5\right)\left(x+9\right)$$

Aromātai

$$\left(x+5\right)\left(x+9\right)$$