Whakaoti i te -18x^2=-27x+4

$$- 18 x ^ { 2 } = - 27 x + 4$$

Whakaoti mō x

$$x=\frac{1}{6}\approx 0.166666667$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 27x ki ngā taha e rua.

$$-18x^{2}+27x=4$$

Tangohia te 4 mai i ngā taha e rua.

$$-18x^{2}+27x-4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -18x^{2}+ax+bx-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=27 ab=-18\left(-4\right)=72$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. 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.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

Ko te otinga te takirua ka hoatu i te tapeke 27.

$$a=24 b=3$$

Tuhia anō te -18x^{2}+27x-4 hei \left(-18x^{2}+24x\right)+\left(3x-4\right).

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

Whakatauwehea atu -6x i te -18x^{2}+24x.

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

Whakatauwehea atu te kīanga pātahi 3x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-4\right)\left(-6x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-4=0 me te -6x+1=0.

$$x=\frac{4}{3} x=\frac{1}{6}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 27x ki ngā taha e rua.

$$-18x^{2}+27x=4$$

Tangohia te 4 mai i ngā taha e rua.

$$-18x^{2}+27x-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -18 mō a, 27 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-18\right)\left(-4\right)}}{2\left(-18\right)}$$

Pūrua 27.

$$x=\frac{-27±\sqrt{729-4\left(-18\right)\left(-4\right)}}{2\left(-18\right)}$$

Whakareatia -4 ki te -18.

$$x=\frac{-27±\sqrt{729+72\left(-4\right)}}{2\left(-18\right)}$$

Whakareatia 72 ki te -4.

$$x=\frac{-27±\sqrt{729-288}}{2\left(-18\right)}$$

Tāpiri 729 ki te -288.

$$x=\frac{-27±\sqrt{441}}{2\left(-18\right)}$$

Tuhia te pūtakerua o te 441.

$$x=\frac{-27±21}{2\left(-18\right)}$$

Whakareatia 2 ki te -18.

$$x=\frac{-27±21}{-36}$$

Nā, me whakaoti te whārite x=\frac{-27±21}{-36} ina he tāpiri te ±. Tāpiri -27 ki te 21.

$$x=-\frac{6}{-36}$$

Whakahekea te hautanga \frac{-6}{-36} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{1}{6}$$

Nā, me whakaoti te whārite x=\frac{-27±21}{-36} ina he tango te ±. Tango 21 mai i -27.

$$x=-\frac{48}{-36}$$

Whakahekea te hautanga \frac{-48}{-36} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

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

Kua oti te whārite te whakatau.

$$x=\frac{1}{6} x=\frac{4}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 27x ki ngā taha e rua.

$$-18x^{2}+27x=4$$

Whakawehea ngā taha e rua ki te -18.

$$\frac{-18x^{2}+27x}{-18}=\frac{4}{-18}$$

Mā te whakawehe ki te -18 ka wetekia te whakareanga ki te -18.

$$x^{2}+\frac{27}{-18}x=\frac{4}{-18}$$

Whakahekea te hautanga \frac{27}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 9.

$$x^{2}-\frac{3}{2}x=\frac{4}{-18}$$

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

$$x^{2}-\frac{3}{2}x=-\frac{2}{9}$$

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.

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

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

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

Tāpiri -\frac{2}{9} ki te \frac{9}{16} 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.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{49}{144}$$

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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{49}{144}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{7}{12} x-\frac{3}{4}=-\frac{7}{12}$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

$$x=\frac{4}{3} x=\frac{1}{6}$$