Whakaoti i te 30+13x-3x^2

$$30 + 13 x - 3 x ^ { 2 }$$

Tauwehe

$$\left(6-x\right)\left(3x+5\right)$$

Ngā Upane Otinga

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-3x^{2}+13x+30$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -3x^{2}+ax+bx+30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=-3\times 30=-90$$

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 -90.

$$-1,90 -2,45 -3,30 -5,18 -6,15 -9,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+90=89 -2+45=43 -3+30=27 -5+18=13 -6+15=9 -9+10=1$$

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

$$a=18 b=-5$$

Tuhia anō te -3x^{2}+13x+30 hei \left(-3x^{2}+18x\right)+\left(-5x+30\right).

$$\left(-3x^{2}+18x\right)+\left(-5x+30\right)$$

Tauwehea te 3x i te tuatahi me te 5 i te rōpū tuarua.

$$3x\left(-x+6\right)+5\left(-x+6\right)$$

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

$$\left(-x+6\right)\left(3x+5\right)$$

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.

$$-3x^{2}+13x+30=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-3\right)\times 30}}{2\left(-3\right)}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\left(-3\right)\times 30}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-13±\sqrt{169+12\times 30}}{2\left(-3\right)}$$

Whakareatia 12 ki te 30.

$$x=\frac{-13±\sqrt{169+360}}{2\left(-3\right)}$$

Tāpiri 169 ki te 360.

$$x=\frac{-13±\sqrt{529}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-13±23}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-13±23}{-6}$$

Nā, me whakaoti te whārite x=\frac{-13±23}{-6} ina he tāpiri te ±. Tāpiri -13 ki te 23.

$$x=\frac{10}{-6}$$

Whakahekea te hautanga \frac{10}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{5}{3}$$

Nā, me whakaoti te whārite x=\frac{-13±23}{-6} ina he tango te ±. Tango 23 mai i -13.

$$x=-\frac{36}{-6}$$

Whakawehe -36 ki te -6.

$$x=6$$

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{5}{3} mō te x_{1} me te 6 mō te x_{2}.

$$-3x^{2}+13x+30=-3\left(x-\left(-\frac{5}{3}\right)\right)\left(x-6\right)$$

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

$$-3x^{2}+13x+30=-3\left(x+\frac{5}{3}\right)\left(x-6\right)$$

Tāpiri \frac{5}{3} 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.

$$-3x^{2}+13x+30=-3\times \frac{-3x-5}{-3}\left(x-6\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te -3 me te 3.

$$-3x^{2}+13x+30=\left(-3x-5\right)\left(x-6\right)$$

Aromātai

$$\left(6-x\right)\left(3x+5\right)$$