Whakaoti i te 4x^2+17x-42

$$4 x ^ { 2 } + 17 x - 42$$

Tauwehe

$$\left(4x-7\right)\left(x+6\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4x^{2}+ax+bx-42. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=4\left(-42\right)=-168$$

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 -168.

$$-1,168 -2,84 -3,56 -4,42 -6,28 -7,24 -8,21 -12,14$$

Tātaihia te tapeke mō ia takirua.

$$-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2$$

Ko te otinga te takirua ka hoatu i te tapeke 17.

$$a=-7 b=24$$

Tuhia anō te 4x^{2}+17x-42 hei \left(4x^{2}-7x\right)+\left(24x-42\right).

$$\left(4x^{2}-7x\right)+\left(24x-42\right)$$

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

$$x\left(4x-7\right)+6\left(4x-7\right)$$

Whakatauwehea atu te kīanga pātahi 4x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-7\right)\left(x+6\right)$$

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.

$$4x^{2}+17x-42=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\times 4\left(-42\right)}}{2\times 4}$$

Pūrua 17.

$$x=\frac{-17±\sqrt{289-4\times 4\left(-42\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-17±\sqrt{289-16\left(-42\right)}}{2\times 4}$$

Whakareatia -16 ki te -42.

$$x=\frac{-17±\sqrt{289+672}}{2\times 4}$$

Tāpiri 289 ki te 672.

$$x=\frac{-17±\sqrt{961}}{2\times 4}$$

Tuhia te pūtakerua o te 961.

$$x=\frac{-17±31}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-17±31}{8}$$

Nā, me whakaoti te whārite x=\frac{-17±31}{8} ina he tāpiri te ±. Tāpiri -17 ki te 31.

$$x=\frac{14}{8}$$

Whakahekea te hautanga \frac{14}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{7}{4}$$

Nā, me whakaoti te whārite x=\frac{-17±31}{8} ina he tango te ±. Tango 31 mai i -17.

$$x=-\frac{48}{8}$$

Whakawehe -48 ki te 8.

$$x=-6$$

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{7}{4} mō te x_{1} me te -6 mō te x_{2}.

$$4x^{2}+17x-42=4\left(x-\frac{7}{4}\right)\left(x-\left(-6\right)\right)$$

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

$$4x^{2}+17x-42=4\left(x-\frac{7}{4}\right)\left(x+6\right)$$

Tango \frac{7}{4} 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.

$$4x^{2}+17x-42=4\times \frac{4x-7}{4}\left(x+6\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te 4 me te 4.

$$4x^{2}+17x-42=\left(4x-7\right)\left(x+6\right)$$

Aromātai

$$\left(4x-7\right)\left(x+6\right)$$