Whakaoti i te 4x^2-21x-18=0

$$4 { x }^{ 2 } -21x-18=0$$

Whakaoti mō x

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

$$x=6$$

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 4x^{2}+ax+bx-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-21 ab=4\left(-18\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -21.

$$a=-24 b=3$$

Tuhia anō te 4x^{2}-21x-18 hei \left(4x^{2}-24x\right)+\left(3x-18\right).

$$\left(4x^{2}-24x\right)+\left(3x-18\right)$$

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

$$4x\left(x-6\right)+3\left(x-6\right)$$

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

$$\left(x-6\right)\left(4x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te 4x+3=0.

$$x=6 x=-\frac{3}{4}$$

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.

$$4x^{2}-21x-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -21 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 4\left(-18\right)}}{2\times 4}$$

Pūrua -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 4\left(-18\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-21\right)±\sqrt{441-16\left(-18\right)}}{2\times 4}$$

Whakareatia -16 ki te -18.

$$x=\frac{-\left(-21\right)±\sqrt{441+288}}{2\times 4}$$

Tāpiri 441 ki te 288.

$$x=\frac{-\left(-21\right)±\sqrt{729}}{2\times 4}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-\left(-21\right)±27}{2\times 4}$$

Ko te tauaro o -21 ko 21.

$$x=\frac{21±27}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{21±27}{8}$$

Nā, me whakaoti te whārite x=\frac{21±27}{8} ina he tāpiri te ±. Tāpiri 21 ki te 27.

$$x=\frac{48}{8}$$

Whakawehe 48 ki te 8.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{21±27}{8} ina he tango te ±. Tango 27 mai i 21.

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

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.

$$x=6 x=-\frac{3}{4}$$

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.

$$4x^{2}-21x-18=0$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$4x^{2}-21x-18-\left(-18\right)=-\left(-18\right)$$

Mā te tango i te -18 i a ia ake anō ka toe ko te 0.

$$4x^{2}-21x=-\left(-18\right)$$

Tango -18 mai i 0.

$$4x^{2}-21x=18$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-21x}{4}=\frac{18}{4}$$

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

$$x^{2}-\frac{21}{4}x=\frac{18}{4}$$

Whakahekea te hautanga \frac{18}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{21}{4}x=\frac{9}{2}$$

Whakawehea te -\frac{21}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{21}{8}. Nā, tāpiria te pūrua o te -\frac{21}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{21}{4}x+\left(-\frac{21}{8}\right)^{2}=\frac{9}{2}+\left(-\frac{21}{8}\right)^{2}$$

Pūruatia -\frac{21}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=\frac{9}{2}+\frac{441}{64}$$

Tāpiri \frac{9}{2} ki te \frac{441}{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.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=\frac{729}{64}$$

Tauwehea x^{2}-\frac{21}{4}x+\frac{441}{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}.

$$\left(x-\frac{21}{8}\right)^{2}=\frac{729}{64}$$

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

$$\sqrt{\left(x-\frac{21}{8}\right)^{2}}=\sqrt{\frac{729}{64}}$$

Whakarūnātia.

$$x-\frac{21}{8}=\frac{27}{8} x-\frac{21}{8}=-\frac{27}{8}$$

Me tāpiri \frac{21}{8} ki ngā taha e rua o te whārite.

$$x=6 x=-\frac{3}{4}$$