Whakaoti i te 8x^2+26x+15

$$8 x ^ { 2 } + 26 x + 15$$

Tauwehe

$$\left(2x+5\right)\left(4x+3\right)$$

Ngā Upane Otinga

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

$$a+b=26 ab=8\times 15=120$$

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 120.

$$1,120 2,60 3,40 4,30 5,24 6,20 8,15 10,12$$

Tātaihia te tapeke mō ia takirua.

$$1+120=121 2+60=62 3+40=43 4+30=34 5+24=29 6+20=26 8+15=23 10+12=22$$

Ko te otinga te takirua ka hoatu i te tapeke 26.

$$a=6 b=20$$

Tuhia anō te 8x^{2}+26x+15 hei \left(8x^{2}+6x\right)+\left(20x+15\right).

$$\left(8x^{2}+6x\right)+\left(20x+15\right)$$

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(4x+3\right)+5\left(4x+3\right)$$

Whakatauwehea atu te kīanga pātahi 4x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x+3\right)\left(2x+5\right)$$

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.

$$8x^{2}+26x+15=0$$

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.

$$x=\frac{-26±\sqrt{26^{2}-4\times 8\times 15}}{2\times 8}$$

Pūrua 26.

$$x=\frac{-26±\sqrt{676-4\times 8\times 15}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-26±\sqrt{676-32\times 15}}{2\times 8}$$

Whakareatia -32 ki te 15.

$$x=\frac{-26±\sqrt{676-480}}{2\times 8}$$

Tāpiri 676 ki te -480.

$$x=\frac{-26±\sqrt{196}}{2\times 8}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-26±14}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-26±14}{16}$$

Nā, me whakaoti te whārite x=\frac{-26±14}{16} ina he tāpiri te ±. Tāpiri -26 ki te 14.

$$x=-\frac{12}{16}$$

Whakahekea te hautanga \frac{-12}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{3}{4}$$

Nā, me whakaoti te whārite x=\frac{-26±14}{16} ina he tango te ±. Tango 14 mai i -26.

$$x=-\frac{40}{16}$$

Whakahekea te hautanga \frac{-40}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=-\frac{5}{2}$$

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{3}{4} mō te x_{1} me te -\frac{5}{2} mō te x_{2}.

$$8x^{2}+26x+15=8\left(x-\left(-\frac{3}{4}\right)\right)\left(x-\left(-\frac{5}{2}\right)\right)$$

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

$$8x^{2}+26x+15=8\left(x+\frac{3}{4}\right)\left(x+\frac{5}{2}\right)$$

Tāpiri \frac{3}{4} 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.

$$8x^{2}+26x+15=8\times \frac{4x+3}{4}\left(x+\frac{5}{2}\right)$$

Tāpiri \frac{5}{2} 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.

$$8x^{2}+26x+15=8\times \frac{4x+3}{4}\times \frac{2x+5}{2}$$

Whakareatia \frac{4x+3}{4} ki te \frac{2x+5}{2} 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.

$$8x^{2}+26x+15=8\times \frac{\left(4x+3\right)\left(2x+5\right)}{4\times 2}$$

Whakareatia 4 ki te 2.

$$8x^{2}+26x+15=8\times \frac{\left(4x+3\right)\left(2x+5\right)}{8}$$

Whakakorea atu te tauwehe pūnoa nui rawa 8 i roto i te 8 me te 8.

$$8x^{2}+26x+15=\left(4x+3\right)\left(2x+5\right)$$

Aromātai

$$\left(2x+5\right)\left(4x+3\right)$$