Whakaoti i te 25x^2+25x-150

$$25 x ^ { 2 } + 25 x - 150$$

Tauwehe

$$25\left(x-2\right)\left(x+3\right)$$

Ngā Upane Otinga

Tauwehea te 25.

$$25\left(x^{2}+x-6\right)$$

Whakaarohia te x^{2}+x-6. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=1\left(-6\right)=-6$$

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 -6.

$$-1,6 -2,3$$

Tātaihia te tapeke mō ia takirua.

$$-1+6=5 -2+3=1$$

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

$$a=-2 b=3$$

Tuhia anō te x^{2}+x-6 hei \left(x^{2}-2x\right)+\left(3x-6\right).

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

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

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

$$25\left(x-2\right)\left(x+3\right)$$

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.

$$25x^{2}+25x-150=0$$

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.

$$x=\frac{-25±\sqrt{25^{2}-4\times 25\left(-150\right)}}{2\times 25}$$

Pūrua 25.

$$x=\frac{-25±\sqrt{625-4\times 25\left(-150\right)}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-25±\sqrt{625-100\left(-150\right)}}{2\times 25}$$

Whakareatia -100 ki te -150.

$$x=\frac{-25±\sqrt{625+15000}}{2\times 25}$$

Tāpiri 625 ki te 15000.

$$x=\frac{-25±\sqrt{15625}}{2\times 25}$$

Tuhia te pūtakerua o te 15625.

$$x=\frac{-25±125}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{-25±125}{50}$$

Nā, me whakaoti te whārite x=\frac{-25±125}{50} ina he tāpiri te ±. Tāpiri -25 ki te 125.

$$x=\frac{100}{50}$$

Whakawehe 100 ki te 50.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-25±125}{50} ina he tango te ±. Tango 125 mai i -25.

$$x=-\frac{150}{50}$$

Whakawehe -150 ki te 50.

$$x=-3$$

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 2 mō te x_{1} me te -3 mō te x_{2}.

$$25x^{2}+25x-150=25\left(x-2\right)\left(x-\left(-3\right)\right)$$

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

$$25x^{2}+25x-150=25\left(x-2\right)\left(x+3\right)$$

Aromātai

$$25\left(x-2\right)\left(x+3\right)$$