Whakaoti i te 2x^2+7x-30

$$2 x ^ { 2 } + 7 x - 30$$

Tauwehe

$$\left(2x-5\right)\left(x+6\right)$$

Ngā Upane Otinga

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

$$a+b=7 ab=2\left(-30\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-5 b=12$$

Tuhia anō te 2x^{2}+7x-30 hei \left(2x^{2}-5x\right)+\left(12x-30\right).

$$\left(2x^{2}-5x\right)+\left(12x-30\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(2x-5\right)+6\left(2x-5\right)$$

Whakatauwehea atu te kīanga pātahi 2x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-5\right)\left(x+6\right)$$

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.

$$2x^{2}+7x-30=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 2\left(-30\right)}}{2\times 2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 2\left(-30\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-7±\sqrt{49-8\left(-30\right)}}{2\times 2}$$

Whakareatia -8 ki te -30.

$$x=\frac{-7±\sqrt{49+240}}{2\times 2}$$

Tāpiri 49 ki te 240.

$$x=\frac{-7±\sqrt{289}}{2\times 2}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-7±17}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-7±17}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±17}{4} ina he tāpiri te ±. Tāpiri -7 ki te 17.

$$x=\frac{10}{4}$$

Whakahekea te hautanga \frac{10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-7±17}{4} ina he tango te ±. Tango 17 mai i -7.

$$x=-\frac{24}{4}$$

Whakawehe -24 ki te 4.

$$x=-6$$

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}{2} mō te x_{1} me te -6 mō te x_{2}.

$$2x^{2}+7x-30=2\left(x-\frac{5}{2}\right)\left(x-\left(-6\right)\right)$$

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

$$2x^{2}+7x-30=2\left(x-\frac{5}{2}\right)\left(x+6\right)$$

Tango \frac{5}{2} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$2x^{2}+7x-30=2\times \frac{2x-5}{2}\left(x+6\right)$$

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

$$2x^{2}+7x-30=\left(2x-5\right)\left(x+6\right)$$

Aromātai

$$\left(2x-5\right)\left(x+6\right)$$