Whakaoti i te x^2-6x-55

$$x ^ { 2 } - 6 x - 55$$

Tauwehe

$$\left(x-11\right)\left(x+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-55. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\left(-55\right)=-55$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -55.

$$1,-55 5,-11$$

Tātaihia te tapeke mō ia takirua.

$$1-55=-54 5-11=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-11 b=5$$

Tuhia anō te x^{2}-6x-55 hei \left(x^{2}-11x\right)+\left(5x-55\right).

$$\left(x^{2}-11x\right)+\left(5x-55\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(x-11\right)+5\left(x-11\right)$$

Whakatauwehea atu te kīanga pātahi x-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-11\right)\left(x+5\right)$$

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.

$$x^{2}-6x-55=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-55\right)}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-55\right)}}{2}$$

Whakareatia -4 ki te -55.

$$x=\frac{-\left(-6\right)±\sqrt{36+220}}{2}$$

Tāpiri 36 ki te 220.

$$x=\frac{-\left(-6\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-6\right)±16}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±16}{2}$$

Nā, me whakaoti te whārite x=\frac{6±16}{2} ina he tāpiri te ±. Tāpiri 6 ki te 16.

$$x=\frac{22}{2}$$

Whakawehe 22 ki te 2.

$$x=11$$

Nā, me whakaoti te whārite x=\frac{6±16}{2} ina he tango te ±. Tango 16 mai i 6.

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

Whakawehe -10 ki te 2.

$$x=-5$$

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 11 mō te x_{1} me te -5 mō te x_{2}.

$$x^{2}-6x-55=\left(x-11\right)\left(x-\left(-5\right)\right)$$

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

$$x^{2}-6x-55=\left(x-11\right)\left(x+5\right)$$

Aromātai

$$\left(x-11\right)\left(x+5\right)$$