Whakaoti i te 16x^2-56x+49

$$16 x ^ { 2 } - 56 x + 49$$

Tauwehe

$$\left(4x-7\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 16x^{2}+ax+bx+49. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-56 ab=16\times 49=784$$

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 784.

$$-1,-784 -2,-392 -4,-196 -7,-112 -8,-98 -14,-56 -16,-49 -28,-28$$

Tātaihia te tapeke mō ia takirua.

$$-1-784=-785 -2-392=-394 -4-196=-200 -7-112=-119 -8-98=-106 -14-56=-70 -16-49=-65 -28-28=-56$$

Ko te otinga te takirua ka hoatu i te tapeke -56.

$$a=-28 b=-28$$

Tuhia anō te 16x^{2}-56x+49 hei \left(16x^{2}-28x\right)+\left(-28x+49\right).

$$\left(16x^{2}-28x\right)+\left(-28x+49\right)$$

Tauwehea te 4x i te tuatahi me te -7 i te rōpū tuarua.

$$4x\left(4x-7\right)-7\left(4x-7\right)$$

Whakatauwehea atu te kīanga pātahi 4x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-7\right)\left(4x-7\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(4x-7\right)^{2}$$

Ngā Upane Whakamahi Tauwehe o te Pūrua Huarua

Ko te tikanga tātai o tēnei huatoru he pūrua huatoru, ka whakareatia pea e tētahi tauwehe pātahi. Ka taea ngā pūrua huatoru te tauwehe mā te kimi i ngā pūtakerua o ngā kīanga tau ārahi, autō hoki.

$$factor(16x^{2}-56x+49)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(16,-56,49)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 16x^{2}.

$$\sqrt{16x^{2}}=4x$$

Kimihia te pūtakerua o te kīanga tau autō, 49.

$$\sqrt{49}=7$$

Ko te pūrua huatoru te pūrua o te huarua ko te tapeke tērā, te huatango rānei o ngā pūtakerua o ngā kīanga tau ārahi, autō hoki, e whakaritea ai te tohu e te tohu o te kīanga tau waenga o te pūrua huatoru.

$$\left(4x-7\right)^{2}$$

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.

$$16x^{2}-56x+49=0$$

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.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 16\times 49}}{2\times 16}$$

Pūrua -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 16\times 49}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-\left(-56\right)±\sqrt{3136-64\times 49}}{2\times 16}$$

Whakareatia -64 ki te 49.

$$x=\frac{-\left(-56\right)±\sqrt{3136-3136}}{2\times 16}$$

Tāpiri 3136 ki te -3136.

$$x=\frac{-\left(-56\right)±\sqrt{0}}{2\times 16}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-\left(-56\right)±0}{2\times 16}$$

Ko te tauaro o -56 ko 56.

$$x=\frac{56±0}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{56±0}{32}$$

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{7}{4} mō te x_{1} me te \frac{7}{4} mō te x_{2}.

$$16x^{2}-56x+49=16\left(x-\frac{7}{4}\right)\left(x-\frac{7}{4}\right)$$

Tango \frac{7}{4} 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.

$$16x^{2}-56x+49=16\times \frac{4x-7}{4}\left(x-\frac{7}{4}\right)$$

Tango \frac{7}{4} 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.

$$16x^{2}-56x+49=16\times \frac{4x-7}{4}\times \frac{4x-7}{4}$$

Whakareatia \frac{4x-7}{4} ki te \frac{4x-7}{4} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$16x^{2}-56x+49=16\times \frac{\left(4x-7\right)\left(4x-7\right)}{4\times 4}$$

Whakareatia 4 ki te 4.

$$16x^{2}-56x+49=16\times \frac{\left(4x-7\right)\left(4x-7\right)}{16}$$

Whakakorea atu te tauwehe pūnoa nui rawa 16 i roto i te 16 me te 16.

$$16x^{2}-56x+49=\left(4x-7\right)\left(4x-7\right)$$

Aromātai

$$\left(4x-7\right)^{2}$$