Whakaoti i te 36=3x(-6x+36/2)

$$36 = 3 x ( \frac { - 6 x + 36 } { 2 } )$$

Whakaoti mō x

$$x=\sqrt{5}+3\approx 5.236067977$$

$$x=3-\sqrt{5}\approx 0.763932023$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$72=3x\left(-6x+36\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te -6x+36.

$$72=-18x^{2}+108x$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-18x^{2}+108x=72$$

Tangohia te 72 mai i ngā taha e rua.

$$-18x^{2}+108x-72=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -18 mō a, 108 mō b, me -72 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-108±\sqrt{108^{2}-4\left(-18\right)\left(-72\right)}}{2\left(-18\right)}$$

Pūrua 108.

$$x=\frac{-108±\sqrt{11664-4\left(-18\right)\left(-72\right)}}{2\left(-18\right)}$$

Whakareatia -4 ki te -18.

$$x=\frac{-108±\sqrt{11664+72\left(-72\right)}}{2\left(-18\right)}$$

Whakareatia 72 ki te -72.

$$x=\frac{-108±\sqrt{11664-5184}}{2\left(-18\right)}$$

Tāpiri 11664 ki te -5184.

$$x=\frac{-108±\sqrt{6480}}{2\left(-18\right)}$$

Tuhia te pūtakerua o te 6480.

$$x=\frac{-108±36\sqrt{5}}{2\left(-18\right)}$$

Whakareatia 2 ki te -18.

$$x=\frac{-108±36\sqrt{5}}{-36}$$

Nā, me whakaoti te whārite x=\frac{-108±36\sqrt{5}}{-36} ina he tāpiri te ±. Tāpiri -108 ki te 36\sqrt{5}.

$$x=\frac{36\sqrt{5}-108}{-36}$$

Whakawehe -108+36\sqrt{5} ki te -36.

$$x=3-\sqrt{5}$$

Nā, me whakaoti te whārite x=\frac{-108±36\sqrt{5}}{-36} ina he tango te ±. Tango 36\sqrt{5} mai i -108.

$$x=\frac{-36\sqrt{5}-108}{-36}$$

Whakawehe -108-36\sqrt{5} ki te -36.

$$x=\sqrt{5}+3$$

Kua oti te whārite te whakatau.

$$x=3-\sqrt{5} x=\sqrt{5}+3$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$72=3x\left(-6x+36\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te -6x+36.

$$72=-18x^{2}+108x$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-18x^{2}+108x=72$$

Whakawehea ngā taha e rua ki te -18.

$$\frac{-18x^{2}+108x}{-18}=\frac{72}{-18}$$

Mā te whakawehe ki te -18 ka wetekia te whakareanga ki te -18.

$$x^{2}+\frac{108}{-18}x=\frac{72}{-18}$$

Whakawehe 108 ki te -18.

$$x^{2}-6x=\frac{72}{-18}$$

Whakawehe 72 ki te -18.

$$x^{2}-6x=-4$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=-4+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-4+9$$

Tāpiri -4 ki te 9.

$$x^{2}-6x+9=5$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=5$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$x-3=\sqrt{5} x-3=-\sqrt{5}$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=\sqrt{5}+3 x=3-\sqrt{5}$$