Whakaoti i te 490x^2+560x+160

$$490 { x }^{ 2 } +560x+160$$

Tauwehe

$$10\left(7x+4\right)^{2}$$

Ngā Upane Otinga

Tauwehea te 10.

$$10\left(49x^{2}+56x+16\right)$$

Whakaarohia te 49x^{2}+56x+16. Whakamahia te tikanga tātai pūrua pā, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, ina a=7x, ina b=4.

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

Me tuhi anō te kīanga whakatauwehe katoa.

$$10\left(7x+4\right)^{2}$$

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.

$$factor(490x^{2}+560x+160)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(490,560,160)=10$$

Tauwehea te 10.

$$10\left(49x^{2}+56x+16\right)$$

Kimihia te pūtakerua o te kīanga tau ārahi, 49x^{2}.

$$\sqrt{49x^{2}}=7x$$

Kimihia te pūtakerua o te kīanga tau autō, 16.

$$\sqrt{16}=4$$

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.

$$10\left(7x+4\right)^{2}$$

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.

$$490x^{2}+560x+160=0$$

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.

$$x=\frac{-560±\sqrt{560^{2}-4\times 490\times 160}}{2\times 490}$$

Pūrua 560.

$$x=\frac{-560±\sqrt{313600-4\times 490\times 160}}{2\times 490}$$

Whakareatia -4 ki te 490.

$$x=\frac{-560±\sqrt{313600-1960\times 160}}{2\times 490}$$

Whakareatia -1960 ki te 160.

$$x=\frac{-560±\sqrt{313600-313600}}{2\times 490}$$

Tāpiri 313600 ki te -313600.

$$x=\frac{-560±\sqrt{0}}{2\times 490}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-560±0}{2\times 490}$$

Whakareatia 2 ki te 490.

$$x=\frac{-560±0}{980}$$

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

$$490x^{2}+560x+160=490\left(x-\left(-\frac{4}{7}\right)\right)\left(x-\left(-\frac{4}{7}\right)\right)$$

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

$$490x^{2}+560x+160=490\left(x+\frac{4}{7}\right)\left(x+\frac{4}{7}\right)$$

Tāpiri \frac{4}{7} 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.

$$490x^{2}+560x+160=490\times \frac{7x+4}{7}\left(x+\frac{4}{7}\right)$$

Tāpiri \frac{4}{7} 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.

$$490x^{2}+560x+160=490\times \frac{7x+4}{7}\times \frac{7x+4}{7}$$

Whakareatia \frac{7x+4}{7} ki te \frac{7x+4}{7} 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.

$$490x^{2}+560x+160=490\times \frac{\left(7x+4\right)\left(7x+4\right)}{7\times 7}$$

Whakareatia 7 ki te 7.

$$490x^{2}+560x+160=490\times \frac{\left(7x+4\right)\left(7x+4\right)}{49}$$

Whakakorea atu te tauwehe pūnoa nui rawa 49 i roto i te 490 me te 49.

$$490x^{2}+560x+160=10\left(7x+4\right)\left(7x+4\right)$$

Aromātai

$$10\left(7x+4\right)^{2}$$