Whakaoti i te x^2+5x-14quadtext{2}quad3x^2+20x+25

$$x ^ { 2 } + 5 x - 14 \quad \text { 2 } \quad 3 x ^ { 2 } + 20 x + 25$$

Aromātai

$$25+25x-83x^{2}$$

Ngā Upane Otinga

Whakareatia te 14 ki te 2, ka 28.

$$x^{2}+5x-28\times 3x^{2}+20x+25$$

Whakareatia te 28 ki te 3, ka 84.

$$x^{2}+5x-84x^{2}+20x+25$$

Pahekotia te x^{2} me -84x^{2}, ka -83x^{2}.

$$-83x^{2}+5x+20x+25$$

Pahekotia te 5x me 20x, ka 25x.

$$-83x^{2}+25x+25$$

Tauwehe

$$-83\left(x-\frac{25-5\sqrt{357}}{166}\right)\left(x-\frac{5\sqrt{357}+25}{166}\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 14 ki te 2, ka 28.

$$factor(x^{2}+5x-28\times 3x^{2}+20x+25)$$

Whakareatia te 28 ki te 3, ka 84.

$$factor(x^{2}+5x-84x^{2}+20x+25)$$

Pahekotia te x^{2} me -84x^{2}, ka -83x^{2}.

$$factor(-83x^{2}+5x+20x+25)$$

Pahekotia te 5x me 20x, ka 25x.

$$factor(-83x^{2}+25x+25)$$

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.

$$-83x^{2}+25x+25=0$$

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.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-83\right)\times 25}}{2\left(-83\right)}$$

Pūrua 25.

$$x=\frac{-25±\sqrt{625-4\left(-83\right)\times 25}}{2\left(-83\right)}$$

Whakareatia -4 ki te -83.

$$x=\frac{-25±\sqrt{625+332\times 25}}{2\left(-83\right)}$$

Whakareatia 332 ki te 25.

$$x=\frac{-25±\sqrt{625+8300}}{2\left(-83\right)}$$

Tāpiri 625 ki te 8300.

$$x=\frac{-25±\sqrt{8925}}{2\left(-83\right)}$$

Tuhia te pūtakerua o te 8925.

$$x=\frac{-25±5\sqrt{357}}{2\left(-83\right)}$$

Whakareatia 2 ki te -83.

$$x=\frac{-25±5\sqrt{357}}{-166}$$

Nā, me whakaoti te whārite x=\frac{-25±5\sqrt{357}}{-166} ina he tāpiri te ±. Tāpiri -25 ki te 5\sqrt{357}.

$$x=\frac{5\sqrt{357}-25}{-166}$$

Whakawehe -25+5\sqrt{357} ki te -166.

$$x=\frac{25-5\sqrt{357}}{166}$$

Nā, me whakaoti te whārite x=\frac{-25±5\sqrt{357}}{-166} ina he tango te ±. Tango 5\sqrt{357} mai i -25.

$$x=\frac{-5\sqrt{357}-25}{-166}$$

Whakawehe -25-5\sqrt{357} ki te -166.

$$x=\frac{5\sqrt{357}+25}{166}$$

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{25-5\sqrt{357}}{166} mō te x_{1} me te \frac{25+5\sqrt{357}}{166} mō te x_{2}.

$$-83x^{2}+25x+25=-83\left(x-\frac{25-5\sqrt{357}}{166}\right)\left(x-\frac{5\sqrt{357}+25}{166}\right)$$