Whakaoti i te x^2+19x+78

$$x ^ { 2 } + 19 x + 78$$

Tauwehe

$$\left(x+6\right)\left(x+13\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+78. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=19 ab=1\times 78=78$$

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 78.

$$1,78 2,39 3,26 6,13$$

Tātaihia te tapeke mō ia takirua.

$$1+78=79 2+39=41 3+26=29 6+13=19$$

Ko te otinga te takirua ka hoatu i te tapeke 19.

$$a=6 b=13$$

Tuhia anō te x^{2}+19x+78 hei \left(x^{2}+6x\right)+\left(13x+78\right).

$$\left(x^{2}+6x\right)+\left(13x+78\right)$$

Tauwehea te x i te tuatahi me te 13 i te rōpū tuarua.

$$x\left(x+6\right)+13\left(x+6\right)$$

Whakatauwehea atu te kīanga pātahi x+6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+6\right)\left(x+13\right)$$

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.

$$x^{2}+19x+78=0$$

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.

$$x=\frac{-19±\sqrt{19^{2}-4\times 78}}{2}$$

Pūrua 19.

$$x=\frac{-19±\sqrt{361-4\times 78}}{2}$$

Whakareatia -4 ki te 78.

$$x=\frac{-19±\sqrt{361-312}}{2}$$

Tāpiri 361 ki te -312.

$$x=\frac{-19±\sqrt{49}}{2}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-19±7}{2}$$

Nā, me whakaoti te whārite x=\frac{-19±7}{2} ina he tāpiri te ±. Tāpiri -19 ki te 7.

$$x=-\frac{12}{2}$$

Whakawehe -12 ki te 2.

$$x=-6$$

Nā, me whakaoti te whārite x=\frac{-19±7}{2} ina he tango te ±. Tango 7 mai i -19.

$$x=-\frac{26}{2}$$

Whakawehe -26 ki te 2.

$$x=-13$$

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 -6 mō te x_{1} me te -13 mō te x_{2}.

$$x^{2}+19x+78=\left(x-\left(-6\right)\right)\left(x-\left(-13\right)\right)$$

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

$$x^{2}+19x+78=\left(x+6\right)\left(x+13\right)$$

Aromātai

$$\left(x+6\right)\left(x+13\right)$$