Whakaoti i te 6x^2-3x-45

$$6 x ^ { 2 } - 3 x - 45$$

Tauwehe

$$3\left(x-3\right)\left(2x+5\right)$$

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(2x^{2}-x-15\right)$$

Whakaarohia te 2x^{2}-x-15. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=2\left(-15\right)=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-6 b=5$$

Tuhia anō te 2x^{2}-x-15 hei \left(2x^{2}-6x\right)+\left(5x-15\right).

$$\left(2x^{2}-6x\right)+\left(5x-15\right)$$

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(x-3\right)+5\left(x-3\right)$$

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-3\right)\left(2x+5\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$3\left(x-3\right)\left(2x+5\right)$$

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.

$$6x^{2}-3x-45=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 6\left(-45\right)}}{2\times 6}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 6\left(-45\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-3\right)±\sqrt{9-24\left(-45\right)}}{2\times 6}$$

Whakareatia -24 ki te -45.

$$x=\frac{-\left(-3\right)±\sqrt{9+1080}}{2\times 6}$$

Tāpiri 9 ki te 1080.

$$x=\frac{-\left(-3\right)±\sqrt{1089}}{2\times 6}$$

Tuhia te pūtakerua o te 1089.

$$x=\frac{-\left(-3\right)±33}{2\times 6}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±33}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{3±33}{12}$$

Nā, me whakaoti te whārite x=\frac{3±33}{12} ina he tāpiri te ±. Tāpiri 3 ki te 33.

$$x=\frac{36}{12}$$

Whakawehe 36 ki te 12.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{3±33}{12} ina he tango te ±. Tango 33 mai i 3.

$$x=-\frac{30}{12}$$

Whakahekea te hautanga \frac{-30}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{5}{2}$$

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 3 mō te x_{1} me te -\frac{5}{2} mō te x_{2}.

$$6x^{2}-3x-45=6\left(x-3\right)\left(x-\left(-\frac{5}{2}\right)\right)$$

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

$$6x^{2}-3x-45=6\left(x-3\right)\left(x+\frac{5}{2}\right)$$

Tāpiri \frac{5}{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.

$$6x^{2}-3x-45=6\left(x-3\right)\times \frac{2x+5}{2}$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 6 me te 2.

$$6x^{2}-3x-45=3\left(x-3\right)\left(2x+5\right)$$

Aromātai

$$3\left(x-3\right)\left(2x+5\right)$$