Whakaoti i te 40x+2x(30-2x)=200

$$40x+2x(30-2x)=200$$

Whakaoti mō x

$$x = \frac{5 \sqrt{17} + 25}{2} \approx 22.807764064$$

$$x = \frac{25 - 5 \sqrt{17}}{2} \approx 2.192235936$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 30-2x.

$$40x+60x-4x^{2}=200$$

Pahekotia te 40x me 60x, ka 100x.

$$100x-4x^{2}=200$$

Tangohia te 200 mai i ngā taha e rua.

$$100x-4x^{2}-200=0$$

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.

$$-4x^{2}+100x-200=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, 100 mō b, me -200 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-4\right)\left(-200\right)}}{2\left(-4\right)}$$

Pūrua 100.

$$x=\frac{-100±\sqrt{10000-4\left(-4\right)\left(-200\right)}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$x=\frac{-100±\sqrt{10000+16\left(-200\right)}}{2\left(-4\right)}$$

Whakareatia 16 ki te -200.

$$x=\frac{-100±\sqrt{10000-3200}}{2\left(-4\right)}$$

Tāpiri 10000 ki te -3200.

$$x=\frac{-100±\sqrt{6800}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te 6800.

$$x=\frac{-100±20\sqrt{17}}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$x=\frac{-100±20\sqrt{17}}{-8}$$

Nā, me whakaoti te whārite x=\frac{-100±20\sqrt{17}}{-8} ina he tāpiri te ±. Tāpiri -100 ki te 20\sqrt{17}.

$$x=\frac{20\sqrt{17}-100}{-8}$$

Whakawehe -100+20\sqrt{17} ki te -8.

$$x=\frac{25-5\sqrt{17}}{2}$$

Nā, me whakaoti te whārite x=\frac{-100±20\sqrt{17}}{-8} ina he tango te ±. Tango 20\sqrt{17} mai i -100.

$$x=\frac{-20\sqrt{17}-100}{-8}$$

Whakawehe -100-20\sqrt{17} ki te -8.

$$x=\frac{5\sqrt{17}+25}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{25-5\sqrt{17}}{2} x=\frac{5\sqrt{17}+25}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 30-2x.

$$40x+60x-4x^{2}=200$$

Pahekotia te 40x me 60x, ka 100x.

$$100x-4x^{2}=200$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-4x^{2}+100x=200$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4x^{2}+100x}{-4}=\frac{200}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$x^{2}+\frac{100}{-4}x=\frac{200}{-4}$$

Whakawehe 100 ki te -4.

$$x^{2}-25x=\frac{200}{-4}$$

Whakawehe 200 ki te -4.

$$x^{2}-25x=-50$$

Whakawehea te -25, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{25}{2}. Nā, tāpiria te pūrua o te -\frac{25}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-50+\left(-\frac{25}{2}\right)^{2}$$

Pūruatia -\frac{25}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-25x+\frac{625}{4}=-50+\frac{625}{4}$$

Tāpiri -50 ki te \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{425}{4}$$

Tauwehea x^{2}-25x+\frac{625}{4}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{425}{4}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{425}{4}}$$

Whakarūnātia.

$$x-\frac{25}{2}=\frac{5\sqrt{17}}{2} x-\frac{25}{2}=-\frac{5\sqrt{17}}{2}$$

Me tāpiri \frac{25}{2} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{17}+25}{2} x=\frac{25-5\sqrt{17}}{2}$$