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

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

Tauwehe

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

Ngā Upane Otinga

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

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

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

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

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

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

Whakatauwehea atu te kīanga pātahi -x+7 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

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

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.

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

Pūrua 13.

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

Whakareatia -4 ki te -2.

$$x=\frac{-13±\sqrt{169+8\times 7}}{2\left(-2\right)}$$

Whakareatia 8 ki te 7.

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

Tāpiri 169 ki te 56.

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

Tuhia te pūtakerua o te 225.

$$x=\frac{-13±15}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-13±15}{-4}$$

Nā, me whakaoti te whārite x=\frac{-13±15}{-4} ina he tāpiri te ±. Tāpiri -13 ki te 15.

$$x=\frac{2}{-4}$$

Whakahekea te hautanga \frac{2}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-13±15}{-4} ina he tango te ±. Tango 15 mai i -13.

$$x=-\frac{28}{-4}$$

Whakawehe -28 ki te -4.

$$x=7$$

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}{2} mō te x_{1} me te 7 mō te x_{2}.

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

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

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

Tāpiri \frac{1}{2} 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.

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

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te -2 me te 2.

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

Aromātai

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