Whakaoti i te 36x^2+60x+25

$$36 { x }^{ 2 } +60x+25$$

Tauwehe

$$\left(6x+5\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 36x^{2}+ax+bx+25. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=60 ab=36\times 25=900$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 900.

$$1,900 2,450 3,300 4,225 5,180 6,150 9,100 10,90 12,75 15,60 18,50 20,45 25,36 30,30$$

Tātaihia te tapeke mō ia takirua.

$$1+900=901 2+450=452 3+300=303 4+225=229 5+180=185 6+150=156 9+100=109 10+90=100 12+75=87 15+60=75 18+50=68 20+45=65 25+36=61 30+30=60$$

Ko te otinga te takirua ka hoatu i te tapeke 60.

$$a=30 b=30$$

Tuhia anō te 36x^{2}+60x+25 hei \left(36x^{2}+30x\right)+\left(30x+25\right).

$$\left(36x^{2}+30x\right)+\left(30x+25\right)$$

Tauwehea te 6x i te tuatahi me te 5 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 6x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Tuhia anōtia hei pūrua huarua.

$$\left(6x+5\right)^{2}$$

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.

$$factor(36x^{2}+60x+25)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(36,60,25)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 36x^{2}.

$$\sqrt{36x^{2}}=6x$$

Kimihia te pūtakerua o te kīanga tau autō, 25.

$$\sqrt{25}=5$$

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.

$$\left(6x+5\right)^{2}$$

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.

$$36x^{2}+60x+25=0$$

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.

$$x=\frac{-60±\sqrt{60^{2}-4\times 36\times 25}}{2\times 36}$$

Pūrua 60.

$$x=\frac{-60±\sqrt{3600-4\times 36\times 25}}{2\times 36}$$

Whakareatia -4 ki te 36.

$$x=\frac{-60±\sqrt{3600-144\times 25}}{2\times 36}$$

Whakareatia -144 ki te 25.

$$x=\frac{-60±\sqrt{3600-3600}}{2\times 36}$$

Tāpiri 3600 ki te -3600.

$$x=\frac{-60±\sqrt{0}}{2\times 36}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-60±0}{2\times 36}$$

Whakareatia 2 ki te 36.

$$x=\frac{-60±0}{72}$$

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}{6} mō te x_{1} me te -\frac{5}{6} mō te x_{2}.

$$36x^{2}+60x+25=36\left(x-\left(-\frac{5}{6}\right)\right)\left(x-\left(-\frac{5}{6}\right)\right)$$

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

$$36x^{2}+60x+25=36\left(x+\frac{5}{6}\right)\left(x+\frac{5}{6}\right)$$

Tāpiri \frac{5}{6} 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.

$$36x^{2}+60x+25=36\times \frac{6x+5}{6}\left(x+\frac{5}{6}\right)$$

Tāpiri \frac{5}{6} 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.

$$36x^{2}+60x+25=36\times \frac{6x+5}{6}\times \frac{6x+5}{6}$$

Whakareatia \frac{6x+5}{6} ki te \frac{6x+5}{6} 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.

$$36x^{2}+60x+25=36\times \frac{\left(6x+5\right)\left(6x+5\right)}{6\times 6}$$

Whakareatia 6 ki te 6.

$$36x^{2}+60x+25=36\times \frac{\left(6x+5\right)\left(6x+5\right)}{36}$$

Whakakorea atu te tauwehe pūnoa nui rawa 36 i roto i te 36 me te 36.

$$36x^{2}+60x+25=\left(6x+5\right)\left(6x+5\right)$$

Aromātai

$$\left(6x+5\right)^{2}$$