Whakaoti i te 4x^2-32x+60

$$4 x ^ { 2 } - 32 x + 60$$

Tauwehe

$$4\left(x-5\right)\left(x-3\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(x^{2}-8x+15\right)$$

Whakaarohia te x^{2}-8x+15. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\times 15=15$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 15.

$$-1,-15 -3,-5$$

Tātaihia te tapeke mō ia takirua.

$$-1-15=-16 -3-5=-8$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-5 b=-3$$

Tuhia anō te x^{2}-8x+15 hei \left(x^{2}-5x\right)+\left(-3x+15\right).

$$\left(x^{2}-5x\right)+\left(-3x+15\right)$$

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

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(x-5\right)\left(x-3\right)$$

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.

$$4x^{2}-32x+60=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 4\times 60}}{2\times 4}$$

Pūrua -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 4\times 60}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-32\right)±\sqrt{1024-16\times 60}}{2\times 4}$$

Whakareatia -16 ki te 60.

$$x=\frac{-\left(-32\right)±\sqrt{1024-960}}{2\times 4}$$

Tāpiri 1024 ki te -960.

$$x=\frac{-\left(-32\right)±\sqrt{64}}{2\times 4}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-\left(-32\right)±8}{2\times 4}$$

Ko te tauaro o -32 ko 32.

$$x=\frac{32±8}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{32±8}{8}$$

Nā, me whakaoti te whārite x=\frac{32±8}{8} ina he tāpiri te ±. Tāpiri 32 ki te 8.

$$x=\frac{40}{8}$$

Whakawehe 40 ki te 8.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{32±8}{8} ina he tango te ±. Tango 8 mai i 32.

$$x=\frac{24}{8}$$

Whakawehe 24 ki te 8.

$$x=3$$

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 5 mō te x_{1} me te 3 mō te x_{2}.

$$4x^{2}-32x+60=4\left(x-5\right)\left(x-3\right)$$

Aromātai

$$4\left(x-5\right)\left(x-3\right)$$