Whakaoti i te 18x^2-9x-5=0

$$18 { x }^{ 2 } -9x-5 = 0$$

Whakaoti mō x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x=\frac{5}{6}\approx 0.833333333$$

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 18x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=18\left(-5\right)=-90$$

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 -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

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

$$a=-15 b=6$$

Tuhia anō te 18x^{2}-9x-5 hei \left(18x^{2}-15x\right)+\left(6x-5\right).

$$\left(18x^{2}-15x\right)+\left(6x-5\right)$$

Whakatauwehea atu 3x i te 18x^{2}-15x.

$$3x\left(6x-5\right)+6x-5$$

Whakatauwehea atu te kīanga pātahi 6x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6x-5\right)\left(3x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 6x-5=0 me te 3x+1=0.

$$x=\frac{5}{6} x=-\frac{1}{3}$$

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.

$$18x^{2}-9x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 18 mō a, -9 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 18\left(-5\right)}}{2\times 18}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 18\left(-5\right)}}{2\times 18}$$

Whakareatia -4 ki te 18.

$$x=\frac{-\left(-9\right)±\sqrt{81-72\left(-5\right)}}{2\times 18}$$

Whakareatia -72 ki te -5.

$$x=\frac{-\left(-9\right)±\sqrt{81+360}}{2\times 18}$$

Tāpiri 81 ki te 360.

$$x=\frac{-\left(-9\right)±\sqrt{441}}{2\times 18}$$

Tuhia te pūtakerua o te 441.

$$x=\frac{-\left(-9\right)±21}{2\times 18}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±21}{2\times 18}$$

Whakareatia 2 ki te 18.

$$x=\frac{9±21}{36}$$

Nā, me whakaoti te whārite x=\frac{9±21}{36} ina he tāpiri te ±. Tāpiri 9 ki te 21.

$$x=\frac{30}{36}$$

Whakahekea te hautanga \frac{30}{36} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{5}{6}$$

Nā, me whakaoti te whārite x=\frac{9±21}{36} ina he tango te ±. Tango 21 mai i 9.

$$x=-\frac{12}{36}$$

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

$$x=-\frac{1}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{6} x=-\frac{1}{3}$$

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.

$$18x^{2}-9x-5=0$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$18x^{2}-9x-5-\left(-5\right)=-\left(-5\right)$$

Mā te tango i te -5 i a ia ake anō ka toe ko te 0.

$$18x^{2}-9x=-\left(-5\right)$$

Tango -5 mai i 0.

$$18x^{2}-9x=5$$

Whakawehea ngā taha e rua ki te 18.

$$\frac{18x^{2}-9x}{18}=\frac{5}{18}$$

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

$$x^{2}+\left(-\frac{9}{18}\right)x=\frac{5}{18}$$

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

$$x^{2}-\frac{1}{2}x=\frac{5}{18}$$

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{5}{18}+\left(-\frac{1}{4}\right)^{2}$$

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{5}{18}+\frac{1}{16}$$

Tāpiri \frac{5}{18} ki te \frac{1}{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.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{49}{144}$$

Tauwehea te x^{2}-\frac{1}{2}x+\frac{1}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{49}{144}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{7}{12} x-\frac{1}{4}=-\frac{7}{12}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{5}{6} x=-\frac{1}{3}$$