Whakaoti i te 2x^2-x-15=0

$$2 x ^ { 2 } - x - 15 = 0$$

Whakaoti mō x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x=3$$

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 2x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=2\left(-15\right)=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-6 b=5$$

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

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

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(x-3\right)+5\left(x-3\right)$$

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

$$\left(x-3\right)\left(2x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te 2x+5=0.

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

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.

$$2x^{2}-x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -1 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-15\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-15\right)}}{2\times 2}$$

Whakareatia -8 ki te -15.

$$x=\frac{-\left(-1\right)±\sqrt{1+120}}{2\times 2}$$

Tāpiri 1 ki te 120.

$$x=\frac{-\left(-1\right)±\sqrt{121}}{2\times 2}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-\left(-1\right)±11}{2\times 2}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±11}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{1±11}{4}$$

Nā, me whakaoti te whārite x=\frac{1±11}{4} ina he tāpiri te ±. Tāpiri 1 ki te 11.

$$x=\frac{12}{4}$$

Whakawehe 12 ki te 4.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{1±11}{4} ina he tango te ±. Tango 11 mai i 1.

$$x=-\frac{10}{4}$$

Whakahekea te hautanga \frac{-10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{5}{2}$$

Kua oti te whārite te whakatau.

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

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.

$$2x^{2}-x-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$2x^{2}-x-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$2x^{2}-x=-\left(-15\right)$$

Tango -15 mai i 0.

$$2x^{2}-x=15$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-x}{2}=\frac{15}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}-\frac{1}{2}x=\frac{15}{2}$$

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.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{15}{2}+\left(-\frac{1}{4}\right)^{2}$$

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

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{15}{2}+\frac{1}{16}$$

Tāpiri \frac{15}{2} 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.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{121}{16}$$

Tauwehea x^{2}-\frac{1}{2}x+\frac{1}{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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{121}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{11}{4} x-\frac{1}{4}=-\frac{11}{4}$$

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

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