Whakaoti i te (32-2x)x=120

$$(32-2x)x=120$$

Whakaoti mō x

$$x=6$$

$$x=10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 32-2x ki te x.

$$32x-2x^{2}=120$$

Tangohia te 120 mai i ngā taha e rua.

$$32x-2x^{2}-120=0$$

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.

$$-2x^{2}+32x-120=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 32 mō b, me -120 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-2\right)\left(-120\right)}}{2\left(-2\right)}$$

Pūrua 32.

$$x=\frac{-32±\sqrt{1024-4\left(-2\right)\left(-120\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-32±\sqrt{1024+8\left(-120\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -120.

$$x=\frac{-32±\sqrt{1024-960}}{2\left(-2\right)}$$

Tāpiri 1024 ki te -960.

$$x=\frac{-32±\sqrt{64}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-32±8}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-32±8}{-4}$$

Nā, me whakaoti te whārite x=\frac{-32±8}{-4} ina he tāpiri te ±. Tāpiri -32 ki te 8.

$$x=-\frac{24}{-4}$$

Whakawehe -24 ki te -4.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{-32±8}{-4} ina he tango te ±. Tango 8 mai i -32.

$$x=-\frac{40}{-4}$$

Whakawehe -40 ki te -4.

$$x=10$$

Kua oti te whārite te whakatau.

$$x=6 x=10$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 32-2x ki te x.

$$32x-2x^{2}=120$$

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.

$$-2x^{2}+32x=120$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+32x}{-2}=\frac{120}{-2}$$

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

$$x^{2}+\frac{32}{-2}x=\frac{120}{-2}$$

Whakawehe 32 ki te -2.

$$x^{2}-16x=\frac{120}{-2}$$

Whakawehe 120 ki te -2.

$$x^{2}-16x=-60$$

Whakawehea te -16, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -8. Nā, tāpiria te pūrua o te -8 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-16x+\left(-8\right)^{2}=-60+\left(-8\right)^{2}$$

Pūrua -8.

$$x^{2}-16x+64=-60+64$$

Tāpiri -60 ki te 64.

$$x^{2}-16x+64=4$$

Tauwehea x^{2}-16x+64. 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}.

$$\left(x-8\right)^{2}=4$$

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

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-8=2 x-8=-2$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$x=10 x=6$$