Whakaoti i te (40*25)-(left(40-xright)left(25-xright))=1156

$$(40 \times 25)-( \left( 40-x \right) \left( 25-x \right) )=1156$$

Whakaoti mō x (complex solution)

$$x=\frac{65+\sqrt{399}i}{2}\approx 32.5+9.987492178i$$

$$x=\frac{-\sqrt{399}i+65}{2}\approx 32.5-9.987492178i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 40 ki te 25, ka 1000.

$$1000-\left(40-x\right)\left(25-x\right)=1156$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 25-x ka whakakotahi i ngā kupu rite.

$$1000-\left(1000-65x+x^{2}\right)=1156$$

Hei kimi i te tauaro o 1000-65x+x^{2}, kimihia te tauaro o ia taurangi.

$$1000-1000+65x-x^{2}=1156$$

Tangohia te 1000 i te 1000, ka 0.

$$65x-x^{2}=1156$$

Tangohia te 1156 mai i ngā taha e rua.

$$65x-x^{2}-1156=0$$

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.

$$-x^{2}+65x-1156=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 65 mō b, me -1156 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65±\sqrt{65^{2}-4\left(-1\right)\left(-1156\right)}}{2\left(-1\right)}$$

Pūrua 65.

$$x=\frac{-65±\sqrt{4225-4\left(-1\right)\left(-1156\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-65±\sqrt{4225+4\left(-1156\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -1156.

$$x=\frac{-65±\sqrt{4225-4624}}{2\left(-1\right)}$$

Tāpiri 4225 ki te -4624.

$$x=\frac{-65±\sqrt{-399}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -399.

$$x=\frac{-65±\sqrt{399}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-65±\sqrt{399}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-65±\sqrt{399}i}{-2} ina he tāpiri te ±. Tāpiri -65 ki te i\sqrt{399}.

$$x=\frac{-65+\sqrt{399}i}{-2}$$

Whakawehe -65+i\sqrt{399} ki te -2.

$$x=\frac{-\sqrt{399}i+65}{2}$$

Nā, me whakaoti te whārite x=\frac{-65±\sqrt{399}i}{-2} ina he tango te ±. Tango i\sqrt{399} mai i -65.

$$x=\frac{-\sqrt{399}i-65}{-2}$$

Whakawehe -65-i\sqrt{399} ki te -2.

$$x=\frac{65+\sqrt{399}i}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{399}i+65}{2} x=\frac{65+\sqrt{399}i}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 40 ki te 25, ka 1000.

$$1000-\left(40-x\right)\left(25-x\right)=1156$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 25-x ka whakakotahi i ngā kupu rite.

$$1000-\left(1000-65x+x^{2}\right)=1156$$

Hei kimi i te tauaro o 1000-65x+x^{2}, kimihia te tauaro o ia taurangi.

$$1000-1000+65x-x^{2}=1156$$

Tangohia te 1000 i te 1000, ka 0.

$$65x-x^{2}=1156$$

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.

$$-x^{2}+65x=1156$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+65x}{-1}=\frac{1156}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{65}{-1}x=\frac{1156}{-1}$$

Whakawehe 65 ki te -1.

$$x^{2}-65x=\frac{1156}{-1}$$

Whakawehe 1156 ki te -1.

$$x^{2}-65x=-1156$$

Whakawehea te -65, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{65}{2}. Nā, tāpiria te pūrua o te -\frac{65}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-65x+\left(-\frac{65}{2}\right)^{2}=-1156+\left(-\frac{65}{2}\right)^{2}$$

Pūruatia -\frac{65}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-65x+\frac{4225}{4}=-1156+\frac{4225}{4}$$

Tāpiri -1156 ki te \frac{4225}{4}.

$$x^{2}-65x+\frac{4225}{4}=-\frac{399}{4}$$

Tauwehea x^{2}-65x+\frac{4225}{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}.

$$\left(x-\frac{65}{2}\right)^{2}=-\frac{399}{4}$$

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

$$\sqrt{\left(x-\frac{65}{2}\right)^{2}}=\sqrt{-\frac{399}{4}}$$

Whakarūnātia.

$$x-\frac{65}{2}=\frac{\sqrt{399}i}{2} x-\frac{65}{2}=-\frac{\sqrt{399}i}{2}$$

Me tāpiri \frac{65}{2} ki ngā taha e rua o te whārite.

$$x=\frac{65+\sqrt{399}i}{2} x=\frac{-\sqrt{399}i+65}{2}$$