Whakaoti i te U(x)=-x^2+15x-14

$$U ( x ) = - x ^ { 2 } + 15 x - 14$$

Tauwehe

$$\left(1-x\right)\left(x-14\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx-14. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=15 ab=-\left(-14\right)=14$$

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 14.

$$1,14 2,7$$

Tātaihia te tapeke mō ia takirua.

$$1+14=15 2+7=9$$

Ko te otinga te takirua ka hoatu i te tapeke 15.

$$a=14 b=1$$

Tuhia anō te -x^{2}+15x-14 hei \left(-x^{2}+14x\right)+\left(x-14\right).

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

Whakatauwehea atu -x i te -x^{2}+14x.

$$-x\left(x-14\right)+x-14$$

Whakatauwehea atu te kīanga pātahi x-14 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-14\right)\left(-x+1\right)$$

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.

$$-x^{2}+15x-14=0$$

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.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Pūrua 15.

$$x=\frac{-15±\sqrt{225-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-15±\sqrt{225+4\left(-14\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -14.

$$x=\frac{-15±\sqrt{225-56}}{2\left(-1\right)}$$

Tāpiri 225 ki te -56.

$$x=\frac{-15±\sqrt{169}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-15±13}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-15±13}{-2}$$

Nā, me whakaoti te whārite x=\frac{-15±13}{-2} ina he tāpiri te ±. Tāpiri -15 ki te 13.

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

Whakawehe -2 ki te -2.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-15±13}{-2} ina he tango te ±. Tango 13 mai i -15.

$$x=-\frac{28}{-2}$$

Whakawehe -28 ki te -2.

$$x=14$$

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 1 mō te x_{1} me te 14 mō te x_{2}.

$$-x^{2}+15x-14=-\left(x-1\right)\left(x-14\right)$$

Aromātai

$$\left(1-x\right)\left(x-14\right)$$