Whakaoti i te (200-20(x-10)(x-8)=640

$$(200-20(x-10)(x-8)=640$$

Whakaoti mō x (complex solution)

$$x=9+\sqrt{21}i\approx 9+4.582575695i$$

$$x=-\sqrt{21}i+9\approx 9-4.582575695i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 640 mai i ngā taha e rua.

$$200-20\left(x-10\right)\left(x-8\right)-640=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -20 ki te x-10.

$$200+\left(-20x+200\right)\left(x-8\right)-640=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -20x+200 ki te x-8 ka whakakotahi i ngā kupu rite.

$$200-20x^{2}+360x-1600-640=0$$

Tangohia te 1600 i te 200, ka -1400.

$$-1400-20x^{2}+360x-640=0$$

Tangohia te 640 i te -1400, ka -2040.

$$-2040-20x^{2}+360x=0$$

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.

$$-20x^{2}+360x-2040=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -20 mō a, 360 mō b, me -2040 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-360±\sqrt{360^{2}-4\left(-20\right)\left(-2040\right)}}{2\left(-20\right)}$$

Pūrua 360.

$$x=\frac{-360±\sqrt{129600-4\left(-20\right)\left(-2040\right)}}{2\left(-20\right)}$$

Whakareatia -4 ki te -20.

$$x=\frac{-360±\sqrt{129600+80\left(-2040\right)}}{2\left(-20\right)}$$

Whakareatia 80 ki te -2040.

$$x=\frac{-360±\sqrt{129600-163200}}{2\left(-20\right)}$$

Tāpiri 129600 ki te -163200.

$$x=\frac{-360±\sqrt{-33600}}{2\left(-20\right)}$$

Tuhia te pūtakerua o te -33600.

$$x=\frac{-360±40\sqrt{21}i}{2\left(-20\right)}$$

Whakareatia 2 ki te -20.

$$x=\frac{-360±40\sqrt{21}i}{-40}$$

Nā, me whakaoti te whārite x=\frac{-360±40\sqrt{21}i}{-40} ina he tāpiri te ±. Tāpiri -360 ki te 40i\sqrt{21}.

$$x=\frac{-360+40\sqrt{21}i}{-40}$$

Whakawehe -360+40i\sqrt{21} ki te -40.

$$x=-\sqrt{21}i+9$$

Nā, me whakaoti te whārite x=\frac{-360±40\sqrt{21}i}{-40} ina he tango te ±. Tango 40i\sqrt{21} mai i -360.

$$x=\frac{-40\sqrt{21}i-360}{-40}$$

Whakawehe -360-40i\sqrt{21} ki te -40.

$$x=9+\sqrt{21}i$$

Kua oti te whārite te whakatau.

$$x=-\sqrt{21}i+9 x=9+\sqrt{21}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te -1 ki te 20, ka -20.

$$200-20\left(x-10\right)\left(x-8\right)=640$$

Whakamahia te āhuatanga tohatoha hei whakarea te -20 ki te x-10.

$$200+\left(-20x+200\right)\left(x-8\right)=640$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -20x+200 ki te x-8 ka whakakotahi i ngā kupu rite.

$$200-20x^{2}+360x-1600=640$$

Tangohia te 1600 i te 200, ka -1400.

$$-1400-20x^{2}+360x=640$$

Me tāpiri te 1400 ki ngā taha e rua.

$$-20x^{2}+360x=640+1400$$

Tāpirihia te 640 ki te 1400, ka 2040.

$$-20x^{2}+360x=2040$$

Whakawehea ngā taha e rua ki te -20.

$$\frac{-20x^{2}+360x}{-20}=\frac{2040}{-20}$$

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

$$x^{2}+\frac{360}{-20}x=\frac{2040}{-20}$$

Whakawehe 360 ki te -20.

$$x^{2}-18x=\frac{2040}{-20}$$

Whakawehe 2040 ki te -20.

$$x^{2}-18x=-102$$

Whakawehea te -18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -9. Nā, tāpiria te pūrua o te -9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-18x+\left(-9\right)^{2}=-102+\left(-9\right)^{2}$$

Pūrua -9.

$$x^{2}-18x+81=-102+81$$

Tāpiri -102 ki te 81.

$$x^{2}-18x+81=-21$$

Tauwehea x^{2}-18x+81. 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}.

$$\left(x-9\right)^{2}=-21$$

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

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{-21}$$

Whakarūnātia.

$$x-9=\sqrt{21}i x-9=-\sqrt{21}i$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x=9+\sqrt{21}i x=-\sqrt{21}i+9$$