Whakaoti i te -2x^2+2x+9=-5x

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

Whakaoti mō x

$$x=-1$$

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 5x ki ngā taha e rua.

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

Pahekotia te 2x me 5x, ka 7x.

$$-2x^{2}+7x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -2x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=-2\times 9=-18$$

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 -18.

$$-1,18 -2,9 -3,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+18=17 -2+9=7 -3+6=3$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=9 b=-2$$

Tuhia anō te -2x^{2}+7x+9 hei \left(-2x^{2}+9x\right)+\left(-2x+9\right).

$$\left(-2x^{2}+9x\right)+\left(-2x+9\right)$$

Tauwehea te -x i te tuatahi me te -1 i te rōpū tuarua.

$$-x\left(2x-9\right)-\left(2x-9\right)$$

Whakatauwehea atu te kīanga pātahi 2x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-9\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-9=0 me te -x-1=0.

$$x=\frac{9}{2} x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 5x ki ngā taha e rua.

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

Pahekotia te 2x me 5x, ka 7x.

$$-2x^{2}+7x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 7 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-7±\sqrt{49+8\times 9}}{2\left(-2\right)}$$

Whakareatia 8 ki te 9.

$$x=\frac{-7±\sqrt{49+72}}{2\left(-2\right)}$$

Tāpiri 49 ki te 72.

$$x=\frac{-7±\sqrt{121}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-7±11}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-7±11}{-4}$$

Nā, me whakaoti te whārite x=\frac{-7±11}{-4} ina he tāpiri te ±. Tāpiri -7 ki te 11.

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

Whakawehe 4 ki te -4.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-7±11}{-4} ina he tango te ±. Tango 11 mai i -7.

$$x=-\frac{18}{-4}$$

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

$$x=\frac{9}{2}$$

Kua oti te whārite te whakatau.

$$x=-1 x=\frac{9}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 5x ki ngā taha e rua.

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

Pahekotia te 2x me 5x, ka 7x.

$$-2x^{2}+7x+9=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-2x^{2}+7x=-9$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+7x}{-2}=-\frac{9}{-2}$$

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

$$x^{2}+\frac{7}{-2}x=-\frac{9}{-2}$$

Whakawehe 7 ki te -2.

$$x^{2}-\frac{7}{2}x=-\frac{9}{-2}$$

Whakawehe -9 ki te -2.

$$x^{2}-\frac{7}{2}x=\frac{9}{2}$$

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=\frac{9}{2}+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{9}{2}+\frac{49}{16}$$

Tāpiri \frac{9}{2} ki te \frac{49}{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.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{121}{16}$$

Tauwehea x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{121}{16}$$

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

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Whakarūnātia.

$$x-\frac{7}{4}=\frac{11}{4} x-\frac{7}{4}=-\frac{11}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

$$x=\frac{9}{2} x=-1$$