Whakaoti i te -49x^2-4+28x

$$- 49 x ^ { 2 } - 4 + 28 x$$

Tauwehe

$$-\left(7x-2\right)^{2}$$

Ngā Upane Otinga

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-49x^{2}+28x-4$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -49x^{2}+ax+bx-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=28 ab=-49\left(-4\right)=196$$

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 196.

$$1,196 2,98 4,49 7,28 14,14$$

Tātaihia te tapeke mō ia takirua.

$$1+196=197 2+98=100 4+49=53 7+28=35 14+14=28$$

Ko te otinga te takirua ka hoatu i te tapeke 28.

$$a=14 b=14$$

Tuhia anō te -49x^{2}+28x-4 hei \left(-49x^{2}+14x\right)+\left(14x-4\right).

$$\left(-49x^{2}+14x\right)+\left(14x-4\right)$$

Tauwehea te -7x i te tuatahi me te 2 i te rōpū tuarua.

$$-7x\left(7x-2\right)+2\left(7x-2\right)$$

Whakatauwehea atu te kīanga pātahi 7x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-2\right)\left(-7x+2\right)$$

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.

$$-49x^{2}+28x-4=0$$

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.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-49\right)\left(-4\right)}}{2\left(-49\right)}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\left(-49\right)\left(-4\right)}}{2\left(-49\right)}$$

Whakareatia -4 ki te -49.

$$x=\frac{-28±\sqrt{784+196\left(-4\right)}}{2\left(-49\right)}$$

Whakareatia 196 ki te -4.

$$x=\frac{-28±\sqrt{784-784}}{2\left(-49\right)}$$

Tāpiri 784 ki te -784.

$$x=\frac{-28±\sqrt{0}}{2\left(-49\right)}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-28±0}{2\left(-49\right)}$$

Whakareatia 2 ki te -49.

$$x=\frac{-28±0}{-98}$$

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{2}{7} mō te x_{1} me te \frac{2}{7} mō te x_{2}.

$$-49x^{2}+28x-4=-49\left(x-\frac{2}{7}\right)\left(x-\frac{2}{7}\right)$$

Tango \frac{2}{7} 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.

$$-49x^{2}+28x-4=-49\times \frac{-7x+2}{-7}\left(x-\frac{2}{7}\right)$$

Tango \frac{2}{7} 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.

$$-49x^{2}+28x-4=-49\times \frac{-7x+2}{-7}\times \frac{-7x+2}{-7}$$

Whakareatia \frac{-7x+2}{-7} ki te \frac{-7x+2}{-7} 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.

$$-49x^{2}+28x-4=-49\times \frac{\left(-7x+2\right)\left(-7x+2\right)}{-7\left(-7\right)}$$

Whakareatia -7 ki te -7.

$$-49x^{2}+28x-4=-49\times \frac{\left(-7x+2\right)\left(-7x+2\right)}{49}$$

Whakakorea atu te tauwehe pūnoa nui rawa 49 i roto i te -49 me te 49.

$$-49x^{2}+28x-4=-\left(-7x+2\right)\left(-7x+2\right)$$

Aromātai

$$-\left(7x-2\right)^{2}$$