Whakaoti i te -48=-4x^2+4x

$$- 48 = - 4 x ^ { 2 } + 4 x$$

Whakaoti mō x

$$x=-3$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$-4x^{2}+4x=-48$$

Me tāpiri te 48 ki ngā taha e rua.

$$-4x^{2}+4x+48=0$$

Whakawehea ngā taha e rua ki te 4.

$$-x^{2}+x+12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-12=-12$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -12.

$$-1,12 -2,6 -3,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+12=11 -2+6=4 -3+4=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=4 b=-3$$

Tuhia anō te -x^{2}+x+12 hei \left(-x^{2}+4x\right)+\left(-3x+12\right).

$$\left(-x^{2}+4x\right)+\left(-3x+12\right)$$

Tauwehea te -x i te tuatahi me te -3 i te rōpū tuarua.

$$-x\left(x-4\right)-3\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(-x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te -x-3=0.

$$x=4 x=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-4x^{2}+4x=-48$$

Me tāpiri te 48 ki ngā taha e rua.

$$-4x^{2}+4x+48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, 4 mō b, me 48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-4\right)\times 48}}{2\left(-4\right)}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-4\right)\times 48}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$x=\frac{-4±\sqrt{16+16\times 48}}{2\left(-4\right)}$$

Whakareatia 16 ki te 48.

$$x=\frac{-4±\sqrt{16+768}}{2\left(-4\right)}$$

Tāpiri 16 ki te 768.

$$x=\frac{-4±\sqrt{784}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te 784.

$$x=\frac{-4±28}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$x=\frac{-4±28}{-8}$$

Nā, me whakaoti te whārite x=\frac{-4±28}{-8} ina he tāpiri te ±. Tāpiri -4 ki te 28.

$$x=\frac{24}{-8}$$

Whakawehe 24 ki te -8.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-4±28}{-8} ina he tango te ±. Tango 28 mai i -4.

$$x=-\frac{32}{-8}$$

Whakawehe -32 ki te -8.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=-3 x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-4x^{2}+4x=-48$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4x^{2}+4x}{-4}=-\frac{48}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$x^{2}+\frac{4}{-4}x=-\frac{48}{-4}$$

Whakawehe 4 ki te -4.

$$x^{2}-x=-\frac{48}{-4}$$

Whakawehe -48 ki te -4.

$$x^{2}-x=12$$

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=12+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=12+\frac{1}{4}$$

Tāpiri 12 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{49}{4}$$

Tauwehea x^{2}-x+\frac{1}{4}. 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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{7}{2} x-\frac{1}{2}=-\frac{7}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=4 x=-3$$