Whakaoti i te (40-x)(20+2x)=1500

$$(40-x)(20+2x)=1500$$

Whakaoti mō x (complex solution)

$$x=15+5\sqrt{5}i\approx 15+11.180339887i$$

$$x=-5\sqrt{5}i+15\approx 15-11.180339887i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 20+2x ka whakakotahi i ngā kupu rite.

$$800+60x-2x^{2}=1500$$

Tangohia te 1500 mai i ngā taha e rua.

$$800+60x-2x^{2}-1500=0$$

Tangohia te 1500 i te 800, ka -700.

$$-700+60x-2x^{2}=0$$

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.

$$-2x^{2}+60x-700=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 60 mō b, me -700 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-2\right)\left(-700\right)}}{2\left(-2\right)}$$

Pūrua 60.

$$x=\frac{-60±\sqrt{3600-4\left(-2\right)\left(-700\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-60±\sqrt{3600+8\left(-700\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -700.

$$x=\frac{-60±\sqrt{3600-5600}}{2\left(-2\right)}$$

Tāpiri 3600 ki te -5600.

$$x=\frac{-60±\sqrt{-2000}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te -2000.

$$x=\frac{-60±20\sqrt{5}i}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-60±20\sqrt{5}i}{-4}$$

Nā, me whakaoti te whārite x=\frac{-60±20\sqrt{5}i}{-4} ina he tāpiri te ±. Tāpiri -60 ki te 20i\sqrt{5}.

$$x=\frac{-60+20\sqrt{5}i}{-4}$$

Whakawehe -60+20i\sqrt{5} ki te -4.

$$x=-5\sqrt{5}i+15$$

Nā, me whakaoti te whārite x=\frac{-60±20\sqrt{5}i}{-4} ina he tango te ±. Tango 20i\sqrt{5} mai i -60.

$$x=\frac{-20\sqrt{5}i-60}{-4}$$

Whakawehe -60-20i\sqrt{5} ki te -4.

$$x=15+5\sqrt{5}i$$

Kua oti te whārite te whakatau.

$$x=-5\sqrt{5}i+15 x=15+5\sqrt{5}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-x ki te 20+2x ka whakakotahi i ngā kupu rite.

$$800+60x-2x^{2}=1500$$

Tangohia te 800 mai i ngā taha e rua.

$$60x-2x^{2}=1500-800$$

Tangohia te 800 i te 1500, ka 700.

$$60x-2x^{2}=700$$

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.

$$-2x^{2}+60x=700$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+60x}{-2}=\frac{700}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$x^{2}+\frac{60}{-2}x=\frac{700}{-2}$$

Whakawehe 60 ki te -2.

$$x^{2}-30x=\frac{700}{-2}$$

Whakawehe 700 ki te -2.

$$x^{2}-30x=-350$$

Whakawehea te -30, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -15. Nā, tāpiria te pūrua o te -15 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-30x+\left(-15\right)^{2}=-350+\left(-15\right)^{2}$$

Pūrua -15.

$$x^{2}-30x+225=-350+225$$

Tāpiri -350 ki te 225.

$$x^{2}-30x+225=-125$$

Tauwehea x^{2}-30x+225. 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}.

$$\left(x-15\right)^{2}=-125$$

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

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{-125}$$

Whakarūnātia.

$$x-15=5\sqrt{5}i x-15=-5\sqrt{5}i$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$x=15+5\sqrt{5}i x=-5\sqrt{5}i+15$$