Whakaoti i te x^2-19x+48=

$$x ^ { 2 } - 19 x + 48 =$$

Tauwehe

$$\left(x-16\right)\left(x-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+48. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-19 ab=1\times 48=48$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

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

$$a=-16 b=-3$$

Tuhia anō te x^{2}-19x+48 hei \left(x^{2}-16x\right)+\left(-3x+48\right).

$$\left(x^{2}-16x\right)+\left(-3x+48\right)$$

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

$$x\left(x-16\right)-3\left(x-16\right)$$

Whakatauwehea atu te kīanga pātahi x-16 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-16\right)\left(x-3\right)$$

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.

$$x^{2}-19x+48=0$$

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.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 48}}{2}$$

Pūrua -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 48}}{2}$$

Whakareatia -4 ki te 48.

$$x=\frac{-\left(-19\right)±\sqrt{361-192}}{2}$$

Tāpiri 361 ki te -192.

$$x=\frac{-\left(-19\right)±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-19\right)±13}{2}$$

Ko te tauaro o -19 ko 19.

$$x=\frac{19±13}{2}$$

Nā, me whakaoti te whārite x=\frac{19±13}{2} ina he tāpiri te ±. Tāpiri 19 ki te 13.

$$x=\frac{32}{2}$$

Whakawehe 32 ki te 2.

$$x=16$$

Nā, me whakaoti te whārite x=\frac{19±13}{2} ina he tango te ±. Tango 13 mai i 19.

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

Whakawehe 6 ki te 2.

$$x=3$$

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 16 mō te x_{1} me te 3 mō te x_{2}.

$$x^{2}-19x+48=\left(x-16\right)\left(x-3\right)$$

Aromātai

$$\left(x-16\right)\left(x-3\right)$$