Whakaoti i te -7x^2+13x+2

$$- 7 x ^ { 2 } + 13 x + 2$$

Tauwehe

$$\left(2-x\right)\left(7x+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -7x^{2}+ax+bx+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=-7\times 2=-14$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -14.

$$-1,14 -2,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+14=13 -2+7=5$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=14 b=-1$$

Tuhia anō te -7x^{2}+13x+2 hei \left(-7x^{2}+14x\right)+\left(-x+2\right).

$$\left(-7x^{2}+14x\right)+\left(-x+2\right)$$

Whakatauwehea atu 7x i te -7x^{2}+14x.

$$7x\left(-x+2\right)-x+2$$

Whakatauwehea atu te kīanga pātahi -x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+2\right)\left(7x+1\right)$$

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.

$$-7x^{2}+13x+2=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Whakareatia -4 ki te -7.

$$x=\frac{-13±\sqrt{169+28\times 2}}{2\left(-7\right)}$$

Whakareatia 28 ki te 2.

$$x=\frac{-13±\sqrt{169+56}}{2\left(-7\right)}$$

Tāpiri 169 ki te 56.

$$x=\frac{-13±\sqrt{225}}{2\left(-7\right)}$$

Tuhia te pūtakerua o te 225.

$$x=\frac{-13±15}{2\left(-7\right)}$$

Whakareatia 2 ki te -7.

$$x=\frac{-13±15}{-14}$$

Nā, me whakaoti te whārite x=\frac{-13±15}{-14} ina he tāpiri te ±. Tāpiri -13 ki te 15.

$$x=\frac{2}{-14}$$

Whakahekea te hautanga \frac{2}{-14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{1}{7}$$

Nā, me whakaoti te whārite x=\frac{-13±15}{-14} ina he tango te ±. Tango 15 mai i -13.

$$x=-\frac{28}{-14}$$

Whakawehe -28 ki te -14.

$$x=2$$

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 -\frac{1}{7} mō te x_{1} me te 2 mō te x_{2}.

$$-7x^{2}+13x+2=-7\left(x-\left(-\frac{1}{7}\right)\right)\left(x-2\right)$$

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

$$-7x^{2}+13x+2=-7\left(x+\frac{1}{7}\right)\left(x-2\right)$$

Tāpiri \frac{1}{7} ki te x mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-7x^{2}+13x+2=-7\times \frac{-7x-1}{-7}\left(x-2\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 7 i roto i te -7 me te 7.

$$-7x^{2}+13x+2=\left(-7x-1\right)\left(x-2\right)$$

Aromātai

$$\left(2-x\right)\left(7x+1\right)$$