Whakaoti i te 6x^2-7x-5

$$6 x ^ { 2 } - 7 x - 5$$

Tauwehe

$$\left(3x-5\right)\left(2x+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=6\left(-5\right)=-30$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -30.

$$1,-30 2,-15 3,-10 5,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-10 b=3$$

Tuhia anō te 6x^{2}-7x-5 hei \left(6x^{2}-10x\right)+\left(3x-5\right).

$$\left(6x^{2}-10x\right)+\left(3x-5\right)$$

Whakatauwehea atu 2x i te 6x^{2}-10x.

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

Whakatauwehea atu te kīanga pātahi 3x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-5\right)\left(2x+1\right)$$

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.

$$6x^{2}-7x-5=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\left(-5\right)}}{2\times 6}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6\left(-5\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\left(-5\right)}}{2\times 6}$$

Whakareatia -24 ki te -5.

$$x=\frac{-\left(-7\right)±\sqrt{49+120}}{2\times 6}$$

Tāpiri 49 ki te 120.

$$x=\frac{-\left(-7\right)±\sqrt{169}}{2\times 6}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-7\right)±13}{2\times 6}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±13}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{7±13}{12}$$

Nā, me whakaoti te whārite x=\frac{7±13}{12} ina he tāpiri te ±. Tāpiri 7 ki te 13.

$$x=\frac{20}{12}$$

Whakahekea te hautanga \frac{20}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{5}{3}$$

Nā, me whakaoti te whārite x=\frac{7±13}{12} ina he tango te ±. Tango 13 mai i 7.

$$x=-\frac{6}{12}$$

Whakahekea te hautanga \frac{-6}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{1}{2}$$

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

$$6x^{2}-7x-5=6\left(x-\frac{5}{3}\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$6x^{2}-7x-5=6\left(x-\frac{5}{3}\right)\left(x+\frac{1}{2}\right)$$

Tango \frac{5}{3} 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.

$$6x^{2}-7x-5=6\times \frac{3x-5}{3}\left(x+\frac{1}{2}\right)$$

Tāpiri \frac{1}{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.

$$6x^{2}-7x-5=6\times \frac{3x-5}{3}\times \frac{2x+1}{2}$$

Whakareatia \frac{3x-5}{3} ki te \frac{2x+1}{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.

$$6x^{2}-7x-5=6\times \frac{\left(3x-5\right)\left(2x+1\right)}{3\times 2}$$

Whakareatia 3 ki te 2.

$$6x^{2}-7x-5=6\times \frac{\left(3x-5\right)\left(2x+1\right)}{6}$$

Whakakorea atu te tauwehe pūnoa nui rawa 6 i roto i te 6 me te 6.

$$6x^{2}-7x-5=\left(3x-5\right)\left(2x+1\right)$$

Aromātai

$$\left(3x-5\right)\left(2x+1\right)$$