Whakaoti i te 2x^2+4x-576=0

$$2 x ^ { 2 } + 4 x - 576 = 0$$

Whakaoti mō x

$$x=-18$$

$$x=16$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

$$x^{2}+2x-288=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-288. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=1\left(-288\right)=-288$$

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 -288.

$$-1,288 -2,144 -3,96 -4,72 -6,48 -8,36 -9,32 -12,24 -16,18$$

Tātaihia te tapeke mō ia takirua.

$$-1+288=287 -2+144=142 -3+96=93 -4+72=68 -6+48=42 -8+36=28 -9+32=23 -12+24=12 -16+18=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=-16 b=18$$

Tuhia anō te x^{2}+2x-288 hei \left(x^{2}-16x\right)+\left(18x-288\right).

$$\left(x^{2}-16x\right)+\left(18x-288\right)$$

Tauwehea te x i te tuatahi me te 18 i te rōpū tuarua.

$$x\left(x-16\right)+18\left(x-16\right)$$

Whakatauwehea atu te kīanga pātahi x-16 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-16\right)\left(x+18\right)$$

Hei kimi otinga whārite, me whakaoti te x-16=0 me te x+18=0.

$$x=16 x=-18$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$2x^{2}+4x-576=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 4 mō b, me -576 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-576\right)}}{2\times 2}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-576\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-4±\sqrt{16-8\left(-576\right)}}{2\times 2}$$

Whakareatia -8 ki te -576.

$$x=\frac{-4±\sqrt{16+4608}}{2\times 2}$$

Tāpiri 16 ki te 4608.

$$x=\frac{-4±\sqrt{4624}}{2\times 2}$$

Tuhia te pūtakerua o te 4624.

$$x=\frac{-4±68}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-4±68}{4}$$

Nā, me whakaoti te whārite x=\frac{-4±68}{4} ina he tāpiri te ±. Tāpiri -4 ki te 68.

$$x=\frac{64}{4}$$

Whakawehe 64 ki te 4.

$$x=16$$

Nā, me whakaoti te whārite x=\frac{-4±68}{4} ina he tango te ±. Tango 68 mai i -4.

$$x=-\frac{72}{4}$$

Whakawehe -72 ki te 4.

$$x=-18$$

Kua oti te whārite te whakatau.

$$x=16 x=-18$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$2x^{2}+4x-576=0$$

Me tāpiri 576 ki ngā taha e rua o te whārite.

$$2x^{2}+4x-576-\left(-576\right)=-\left(-576\right)$$

Mā te tango i te -576 i a ia ake anō ka toe ko te 0.

$$2x^{2}+4x=-\left(-576\right)$$

Tango -576 mai i 0.

$$2x^{2}+4x=576$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+4x}{2}=\frac{576}{2}$$

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

$$x^{2}+\frac{4}{2}x=\frac{576}{2}$$

Whakawehe 4 ki te 2.

$$x^{2}+2x=\frac{576}{2}$$

Whakawehe 576 ki te 2.

$$x^{2}+2x=288$$

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

$$x^{2}+2x+1^{2}=288+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=288+1$$

Tāpiri 288 ki te 1.

$$x^{2}+2x+1=289$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=289$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{289}$$

Whakarūnātia.

$$x+1=17 x+1=-17$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=16 x=-18$$