Whakaoti i te {l}{6x^2+7x}{-5}

$$\left. \begin{array} { l } { 6 x ^ { 2 } + 7 x } \\ { - 5 } \end{array} \right.$$

Tauwehe

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

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 tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. 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.

$$a=-3 b=10$$

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

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

Tauwehea te 3x i te tuatahi me te 5 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

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

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.

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

Pūrua 7.

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

Whakareatia -4 ki te 6.

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

Whakareatia -24 ki te -5.

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

Tāpiri 49 ki te 120.

$$x=\frac{-7±\sqrt{169}}{2\times 6}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-7±13}{2\times 6}$$

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.

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

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

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

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

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

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

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

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

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

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

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

Tango \frac{1}{2} 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.

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

Tāpiri \frac{5}{3} 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.

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

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

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

Whakareatia 2 ki te 3.

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

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

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

Aromātai

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