Whakaoti i te -2x^2+13x+24=0

$$- 2 x ^ { 2 } + 13 x + 24 = 0$$

Whakaoti mō x

$$x=8$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=-2\times 24=-48$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Tātaihia te tapeke mō ia takirua.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=16 b=-3$$

Tuhia anō te -2x^{2}+13x+24 hei \left(-2x^{2}+16x\right)+\left(-3x+24\right).

$$\left(-2x^{2}+16x\right)+\left(-3x+24\right)$$

Tauwehea te 2x i te tuatahi me te 3 i te rōpū tuarua.

$$2x\left(-x+8\right)+3\left(-x+8\right)$$

Whakatauwehea atu te kīanga pātahi -x+8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+8\right)\left(2x+3\right)$$

Hei kimi otinga whārite, me whakaoti te -x+8=0 me te 2x+3=0.

$$x=8 x=-\frac{3}{2}$$

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.

$$-2x^{2}+13x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 13 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-13±\sqrt{169+8\times 24}}{2\left(-2\right)}$$

Whakareatia 8 ki te 24.

$$x=\frac{-13±\sqrt{169+192}}{2\left(-2\right)}$$

Tāpiri 169 ki te 192.

$$x=\frac{-13±\sqrt{361}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-13±19}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-13±19}{-4}$$

Nā, me whakaoti te whārite x=\frac{-13±19}{-4} ina he tāpiri te ±. Tāpiri -13 ki te 19.

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

Whakahekea te hautanga \frac{6}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-13±19}{-4} ina he tango te ±. Tango 19 mai i -13.

$$x=-\frac{32}{-4}$$

Whakawehe -32 ki te -4.

$$x=8$$

Kua oti te whārite te whakatau.

$$x=-\frac{3}{2} x=8$$

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.

$$-2x^{2}+13x+24=0$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-2x^{2}+13x+24-24=-24$$

Mā te tango i te 24 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+13x=-24$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+13x}{-2}=-\frac{24}{-2}$$

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

$$x^{2}+\frac{13}{-2}x=-\frac{24}{-2}$$

Whakawehe 13 ki te -2.

$$x^{2}-\frac{13}{2}x=-\frac{24}{-2}$$

Whakawehe -24 ki te -2.

$$x^{2}-\frac{13}{2}x=12$$

Whakawehea te -\frac{13}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{4}. Nā, tāpiria te pūrua o te -\frac{13}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=12+\left(-\frac{13}{4}\right)^{2}$$

Pūruatia -\frac{13}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=12+\frac{169}{16}$$

Tāpiri 12 ki te \frac{169}{16}.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{361}{16}$$

Tauwehea x^{2}-\frac{13}{2}x+\frac{169}{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}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{361}{16}$$

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

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{361}{16}}$$

Whakarūnātia.

$$x-\frac{13}{4}=\frac{19}{4} x-\frac{13}{4}=-\frac{19}{4}$$

Me tāpiri \frac{13}{4} ki ngā taha e rua o te whārite.

$$x=8 x=-\frac{3}{2}$$