Whakaoti i te 2x^2+3x-90=0

$$2 x ^ { 2 } + 3 x - 90 = 0$$

Whakaoti mō x

$$x = -\frac{15}{2} = -7\frac{1}{2} = -7.5$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2x^{2}+ax+bx-90. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=2\left(-90\right)=-180$$

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 -180.

$$-1,180 -2,90 -3,60 -4,45 -5,36 -6,30 -9,20 -10,18 -12,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+180=179 -2+90=88 -3+60=57 -4+45=41 -5+36=31 -6+30=24 -9+20=11 -10+18=8 -12+15=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-12 b=15$$

Tuhia anō te 2x^{2}+3x-90 hei \left(2x^{2}-12x\right)+\left(15x-90\right).

$$\left(2x^{2}-12x\right)+\left(15x-90\right)$$

Tauwehea te 2x i te tuatahi me te 15 i te rōpū tuarua.

$$2x\left(x-6\right)+15\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(2x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te 2x+15=0.

$$x=6 x=-\frac{15}{2}$$

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.

$$2x^{2}+3x-90=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 3 mō b, me -90 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\left(-90\right)}}{2\times 2}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\times 2\left(-90\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-3±\sqrt{9-8\left(-90\right)}}{2\times 2}$$

Whakareatia -8 ki te -90.

$$x=\frac{-3±\sqrt{9+720}}{2\times 2}$$

Tāpiri 9 ki te 720.

$$x=\frac{-3±\sqrt{729}}{2\times 2}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-3±27}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-3±27}{4}$$

Nā, me whakaoti te whārite x=\frac{-3±27}{4} ina he tāpiri te ±. Tāpiri -3 ki te 27.

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

Whakawehe 24 ki te 4.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{-3±27}{4} ina he tango te ±. Tango 27 mai i -3.

$$x=-\frac{30}{4}$$

Whakahekea te hautanga \frac{-30}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{15}{2}$$

Kua oti te whārite te whakatau.

$$x=6 x=-\frac{15}{2}$$

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.

$$2x^{2}+3x-90=0$$

Me tāpiri 90 ki ngā taha e rua o te whārite.

$$2x^{2}+3x-90-\left(-90\right)=-\left(-90\right)$$

Mā te tango i te -90 i a ia ake anō ka toe ko te 0.

$$2x^{2}+3x=-\left(-90\right)$$

Tango -90 mai i 0.

$$2x^{2}+3x=90$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+3x}{2}=\frac{90}{2}$$

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

$$x^{2}+\frac{3}{2}x=\frac{90}{2}$$

Whakawehe 90 ki te 2.

$$x^{2}+\frac{3}{2}x=45$$

Whakawehea te \frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{4}. Nā, tāpiria te pūrua o te \frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=45+\left(\frac{3}{4}\right)^{2}$$

Pūruatia \frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=45+\frac{9}{16}$$

Tāpiri 45 ki te \frac{9}{16}.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{729}{16}$$

Tauwehea x^{2}+\frac{3}{2}x+\frac{9}{16}. 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}.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{729}{16}$$

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

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{729}{16}}$$

Whakarūnātia.

$$x+\frac{3}{4}=\frac{27}{4} x+\frac{3}{4}=-\frac{27}{4}$$

Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.

$$x=6 x=-\frac{15}{2}$$