Whakaoti i te (40-x)(20+20x)=1200

$$( 40 - x ) ( 20 + 20 x ) = 1200$$

Whakaoti mō x

$$x = \frac{\sqrt{1441} + 39}{2} \approx 38.480252896$$

$$x=\frac{39-\sqrt{1441}}{2}\approx 0.519747104$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 20+20x ka whakakotahi i ngā kupu rite.

$$800+780x-20x^{2}=1200$$

Tangohia te 1200 mai i ngā taha e rua.

$$800+780x-20x^{2}-1200=0$$

Tangohia te 1200 i te 800, ka -400.

$$-400+780x-20x^{2}=0$$

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.

$$-20x^{2}+780x-400=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -20 mō a, 780 mō b, me -400 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-780±\sqrt{780^{2}-4\left(-20\right)\left(-400\right)}}{2\left(-20\right)}$$

Pūrua 780.

$$x=\frac{-780±\sqrt{608400-4\left(-20\right)\left(-400\right)}}{2\left(-20\right)}$$

Whakareatia -4 ki te -20.

$$x=\frac{-780±\sqrt{608400+80\left(-400\right)}}{2\left(-20\right)}$$

Whakareatia 80 ki te -400.

$$x=\frac{-780±\sqrt{608400-32000}}{2\left(-20\right)}$$

Tāpiri 608400 ki te -32000.

$$x=\frac{-780±\sqrt{576400}}{2\left(-20\right)}$$

Tuhia te pūtakerua o te 576400.

$$x=\frac{-780±20\sqrt{1441}}{2\left(-20\right)}$$

Whakareatia 2 ki te -20.

$$x=\frac{-780±20\sqrt{1441}}{-40}$$

Nā, me whakaoti te whārite x=\frac{-780±20\sqrt{1441}}{-40} ina he tāpiri te ±. Tāpiri -780 ki te 20\sqrt{1441}.

$$x=\frac{20\sqrt{1441}-780}{-40}$$

Whakawehe -780+20\sqrt{1441} ki te -40.

$$x=\frac{39-\sqrt{1441}}{2}$$

Nā, me whakaoti te whārite x=\frac{-780±20\sqrt{1441}}{-40} ina he tango te ±. Tango 20\sqrt{1441} mai i -780.

$$x=\frac{-20\sqrt{1441}-780}{-40}$$

Whakawehe -780-20\sqrt{1441} ki te -40.

$$x=\frac{\sqrt{1441}+39}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{39-\sqrt{1441}}{2} x=\frac{\sqrt{1441}+39}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 20+20x ka whakakotahi i ngā kupu rite.

$$800+780x-20x^{2}=1200$$

Tangohia te 800 mai i ngā taha e rua.

$$780x-20x^{2}=1200-800$$

Tangohia te 800 i te 1200, ka 400.

$$780x-20x^{2}=400$$

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.

$$-20x^{2}+780x=400$$

Whakawehea ngā taha e rua ki te -20.

$$\frac{-20x^{2}+780x}{-20}=\frac{400}{-20}$$

Mā te whakawehe ki te -20 ka wetekia te whakareanga ki te -20.

$$x^{2}+\frac{780}{-20}x=\frac{400}{-20}$$

Whakawehe 780 ki te -20.

$$x^{2}-39x=\frac{400}{-20}$$

Whakawehe 400 ki te -20.

$$x^{2}-39x=-20$$

Whakawehea te -39, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{39}{2}. Nā, tāpiria te pūrua o te -\frac{39}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-39x+\left(-\frac{39}{2}\right)^{2}=-20+\left(-\frac{39}{2}\right)^{2}$$

Pūruatia -\frac{39}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-39x+\frac{1521}{4}=-20+\frac{1521}{4}$$

Tāpiri -20 ki te \frac{1521}{4}.

$$x^{2}-39x+\frac{1521}{4}=\frac{1441}{4}$$

Tauwehea x^{2}-39x+\frac{1521}{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}.

$$\left(x-\frac{39}{2}\right)^{2}=\frac{1441}{4}$$

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

$$\sqrt{\left(x-\frac{39}{2}\right)^{2}}=\sqrt{\frac{1441}{4}}$$

Whakarūnātia.

$$x-\frac{39}{2}=\frac{\sqrt{1441}}{2} x-\frac{39}{2}=-\frac{\sqrt{1441}}{2}$$

Me tāpiri \frac{39}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1441}+39}{2} x=\frac{39-\sqrt{1441}}{2}$$