Whakaoti i te 3x^2-5x-250=0

$$3 { x }^{ 2 } -5x-250=0$$

Whakaoti mō x

$$x = -\frac{25}{3} = -8\frac{1}{3} \approx -8.333333333$$

$$x=10$$

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 3x^{2}+ax+bx-250. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=3\left(-250\right)=-750$$

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 -750.

$$1,-750 2,-375 3,-250 5,-150 6,-125 10,-75 15,-50 25,-30$$

Tātaihia te tapeke mō ia takirua.

$$1-750=-749 2-375=-373 3-250=-247 5-150=-145 6-125=-119 10-75=-65 15-50=-35 25-30=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-30 b=25$$

Tuhia anō te 3x^{2}-5x-250 hei \left(3x^{2}-30x\right)+\left(25x-250\right).

$$\left(3x^{2}-30x\right)+\left(25x-250\right)$$

Tauwehea te 3x i te tuatahi me te 25 i te rōpū tuarua.

$$3x\left(x-10\right)+25\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(3x+25\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te 3x+25=0.

$$x=10 x=-\frac{25}{3}$$

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.

$$3x^{2}-5x-250=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -5 mō b, me -250 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-250\right)}}{2\times 3}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 3\left(-250\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-250\right)}}{2\times 3}$$

Whakareatia -12 ki te -250.

$$x=\frac{-\left(-5\right)±\sqrt{25+3000}}{2\times 3}$$

Tāpiri 25 ki te 3000.

$$x=\frac{-\left(-5\right)±\sqrt{3025}}{2\times 3}$$

Tuhia te pūtakerua o te 3025.

$$x=\frac{-\left(-5\right)±55}{2\times 3}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±55}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{5±55}{6}$$

Nā, me whakaoti te whārite x=\frac{5±55}{6} ina he tāpiri te ±. Tāpiri 5 ki te 55.

$$x=\frac{60}{6}$$

Whakawehe 60 ki te 6.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{5±55}{6} ina he tango te ±. Tango 55 mai i 5.

$$x=-\frac{50}{6}$$

Whakahekea te hautanga \frac{-50}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{25}{3}$$

Kua oti te whārite te whakatau.

$$x=10 x=-\frac{25}{3}$$

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.

$$3x^{2}-5x-250=0$$

Me tāpiri 250 ki ngā taha e rua o te whārite.

$$3x^{2}-5x-250-\left(-250\right)=-\left(-250\right)$$

Mā te tango i te -250 i a ia ake anō ka toe ko te 0.

$$3x^{2}-5x=-\left(-250\right)$$

Tango -250 mai i 0.

$$3x^{2}-5x=250$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-5x}{3}=\frac{250}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{5}{3}x=\frac{250}{3}$$

Whakawehea te -\frac{5}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{6}. Nā, tāpiria te pūrua o te -\frac{5}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=\frac{250}{3}+\left(-\frac{5}{6}\right)^{2}$$

Pūruatia -\frac{5}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{250}{3}+\frac{25}{36}$$

Tāpiri \frac{250}{3} ki te \frac{25}{36} 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.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{3025}{36}$$

Tauwehea x^{2}-\frac{5}{3}x+\frac{25}{36}. 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}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{3025}{36}$$

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

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{3025}{36}}$$

Whakarūnātia.

$$x-\frac{5}{6}=\frac{55}{6} x-\frac{5}{6}=-\frac{55}{6}$$

Me tāpiri \frac{5}{6} ki ngā taha e rua o te whārite.

$$x=10 x=-\frac{25}{3}$$