Whakaoti i te 4x^2-9x-9=0

$$4 x ^ { 2 } - 9 x - 9 = 0$$

Whakaoti mō x

$$x=-\frac{3}{4}=-0.75$$

$$x=3$$

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 4x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=4\left(-9\right)=-36$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-12 b=3$$

Tuhia anō te 4x^{2}-9x-9 hei \left(4x^{2}-12x\right)+\left(3x-9\right).

$$\left(4x^{2}-12x\right)+\left(3x-9\right)$$

Tauwehea te 4x i te tuatahi me te 3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

$$4x^{2}-9x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -9 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\left(-9\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-16\left(-9\right)}}{2\times 4}$$

Whakareatia -16 ki te -9.

$$x=\frac{-\left(-9\right)±\sqrt{81+144}}{2\times 4}$$

Tāpiri 81 ki te 144.

$$x=\frac{-\left(-9\right)±\sqrt{225}}{2\times 4}$$

Tuhia te pūtakerua o te 225.

$$x=\frac{-\left(-9\right)±15}{2\times 4}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±15}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{9±15}{8}$$

Nā, me whakaoti te whārite x=\frac{9±15}{8} ina he tāpiri te ±. Tāpiri 9 ki te 15.

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

Whakawehe 24 ki te 8.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{9±15}{8} ina he tango te ±. Tango 15 mai i 9.

$$x=-\frac{6}{8}$$

Whakahekea te hautanga \frac{-6}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

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.

$$4x^{2}-9x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$4x^{2}-9x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$4x^{2}-9x=-\left(-9\right)$$

Tango -9 mai i 0.

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

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-9x}{4}=\frac{9}{4}$$

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

$$x^{2}-\frac{9}{4}x=\frac{9}{4}$$

Whakawehea te -\frac{9}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{8}. Nā, tāpiria te pūrua o te -\frac{9}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=\frac{9}{4}+\left(-\frac{9}{8}\right)^{2}$$

Pūruatia -\frac{9}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{9}{4}+\frac{81}{64}$$

Tāpiri \frac{9}{4} ki te \frac{81}{64} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{225}{64}$$

Tauwehea x^{2}-\frac{9}{4}x+\frac{81}{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}.

$$\left(x-\frac{9}{8}\right)^{2}=\frac{225}{64}$$

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

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{\frac{225}{64}}$$

Whakarūnātia.

$$x-\frac{9}{8}=\frac{15}{8} x-\frac{9}{8}=-\frac{15}{8}$$

Me tāpiri \frac{9}{8} ki ngā taha e rua o te whārite.

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