Whakaoti i te (22+51x-10x^2)_{max}

$$( 22 + 51 x - 10 x ^ { 2 } ) _ { \max }$$

Tauwehe

$$-\left(2x-11\right)\left(5x+2\right)$$

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.

$$-10x^{2}+51x+22$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -10x^{2}+ax+bx+22. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=51 ab=-10\times 22=-220$$

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 -220.

$$-1,220 -2,110 -4,55 -5,44 -10,22 -11,20$$

Tātaihia te tapeke mō ia takirua.

$$-1+220=219 -2+110=108 -4+55=51 -5+44=39 -10+22=12 -11+20=9$$

Ko te otinga te takirua ka hoatu i te tapeke 51.

$$a=55 b=-4$$

Tuhia anō te -10x^{2}+51x+22 hei \left(-10x^{2}+55x\right)+\left(-4x+22\right).

$$\left(-10x^{2}+55x\right)+\left(-4x+22\right)$$

Tauwehea te -5x i te tuatahi me te -2 i te rōpū tuarua.

$$-5x\left(2x-11\right)-2\left(2x-11\right)$$

Whakatauwehea atu te kīanga pātahi 2x-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-11\right)\left(-5x-2\right)$$

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.

$$-10x^{2}+51x+22=0$$

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.

$$x=\frac{-51±\sqrt{51^{2}-4\left(-10\right)\times 22}}{2\left(-10\right)}$$

Pūrua 51.

$$x=\frac{-51±\sqrt{2601-4\left(-10\right)\times 22}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-51±\sqrt{2601+40\times 22}}{2\left(-10\right)}$$

Whakareatia 40 ki te 22.

$$x=\frac{-51±\sqrt{2601+880}}{2\left(-10\right)}$$

Tāpiri 2601 ki te 880.

$$x=\frac{-51±\sqrt{3481}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te 3481.

$$x=\frac{-51±59}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{-51±59}{-20}$$

Nā, me whakaoti te whārite x=\frac{-51±59}{-20} ina he tāpiri te ±. Tāpiri -51 ki te 59.

$$x=\frac{8}{-20}$$

Whakahekea te hautanga \frac{8}{-20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{2}{5}$$

Nā, me whakaoti te whārite x=\frac{-51±59}{-20} ina he tango te ±. Tango 59 mai i -51.

$$x=-\frac{110}{-20}$$

Whakahekea te hautanga \frac{-110}{-20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{11}{2}$$

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{2}{5} mō te x_{1} me te \frac{11}{2} mō te x_{2}.

$$-10x^{2}+51x+22=-10\left(x-\left(-\frac{2}{5}\right)\right)\left(x-\frac{11}{2}\right)$$

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

$$-10x^{2}+51x+22=-10\left(x+\frac{2}{5}\right)\left(x-\frac{11}{2}\right)$$

Tāpiri \frac{2}{5} 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.

$$-10x^{2}+51x+22=-10\times \frac{-5x-2}{-5}\left(x-\frac{11}{2}\right)$$

Tango \frac{11}{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.

$$-10x^{2}+51x+22=-10\times \frac{-5x-2}{-5}\times \frac{-2x+11}{-2}$$

Whakareatia \frac{-5x-2}{-5} ki te \frac{-2x+11}{-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.

$$-10x^{2}+51x+22=-10\times \frac{\left(-5x-2\right)\left(-2x+11\right)}{-5\left(-2\right)}$$

Whakareatia -5 ki te -2.

$$-10x^{2}+51x+22=-10\times \frac{\left(-5x-2\right)\left(-2x+11\right)}{10}$$

Whakakorea atu te tauwehe pūnoa nui rawa 10 i roto i te -10 me te 10.

$$-10x^{2}+51x+22=-\left(-5x-2\right)\left(-2x+11\right)$$

Aromātai

$$22+51x-10x^{2}$$