Whakaoti i te (x-10)(30-x)=144

$$( x - 10 ) ( 30 - x ) = 144$$

Whakaoti mō x (complex solution)

$$x=-2\sqrt{11}i+20\approx 20-6.633249581i$$

$$x=20+2\sqrt{11}i\approx 20+6.633249581i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-10 ki te 30-x ka whakakotahi i ngā kupu rite.

$$40x-x^{2}-300=144$$

Tangohia te 144 mai i ngā taha e rua.

$$40x-x^{2}-300-144=0$$

Tangohia te 144 i te -300, ka -444.

$$40x-x^{2}-444=0$$

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.

$$-x^{2}+40x-444=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 40 mō b, me -444 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-1\right)\left(-444\right)}}{2\left(-1\right)}$$

Pūrua 40.

$$x=\frac{-40±\sqrt{1600-4\left(-1\right)\left(-444\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-40±\sqrt{1600+4\left(-444\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -444.

$$x=\frac{-40±\sqrt{1600-1776}}{2\left(-1\right)}$$

Tāpiri 1600 ki te -1776.

$$x=\frac{-40±\sqrt{-176}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -176.

$$x=\frac{-40±4\sqrt{11}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-40±4\sqrt{11}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-40±4\sqrt{11}i}{-2} ina he tāpiri te ±. Tāpiri -40 ki te 4i\sqrt{11}.

$$x=\frac{-40+4\sqrt{11}i}{-2}$$

Whakawehe -40+4i\sqrt{11} ki te -2.

$$x=-2\sqrt{11}i+20$$

Nā, me whakaoti te whārite x=\frac{-40±4\sqrt{11}i}{-2} ina he tango te ±. Tango 4i\sqrt{11} mai i -40.

$$x=\frac{-4\sqrt{11}i-40}{-2}$$

Whakawehe -40-4i\sqrt{11} ki te -2.

$$x=20+2\sqrt{11}i$$

Kua oti te whārite te whakatau.

$$x=-2\sqrt{11}i+20 x=20+2\sqrt{11}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-10 ki te 30-x ka whakakotahi i ngā kupu rite.

$$40x-x^{2}-300=144$$

Me tāpiri te 300 ki ngā taha e rua.

$$40x-x^{2}=144+300$$

Tāpirihia te 144 ki te 300, ka 444.

$$40x-x^{2}=444$$

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.

$$-x^{2}+40x=444$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+40x}{-1}=\frac{444}{-1}$$

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

$$x^{2}+\frac{40}{-1}x=\frac{444}{-1}$$

Whakawehe 40 ki te -1.

$$x^{2}-40x=\frac{444}{-1}$$

Whakawehe 444 ki te -1.

$$x^{2}-40x=-444$$

Whakawehea te -40, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -20. Nā, tāpiria te pūrua o te -20 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-40x+\left(-20\right)^{2}=-444+\left(-20\right)^{2}$$

Pūrua -20.

$$x^{2}-40x+400=-444+400$$

Tāpiri -444 ki te 400.

$$x^{2}-40x+400=-44$$

Tauwehea x^{2}-40x+400. 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}.

$$\left(x-20\right)^{2}=-44$$

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

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{-44}$$

Whakarūnātia.

$$x-20=2\sqrt{11}i x-20=-2\sqrt{11}i$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$x=20+2\sqrt{11}i x=-2\sqrt{11}i+20$$